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Theorem omopth2 6763
Description: An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
omopth2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  <-> 
( B  =  D  /\  C  =  E ) ) )

Proof of Theorem omopth2
StepHypRef Expression
1 simpl2l 1010 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  B  e.  On )
2 eloni 4532 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
31, 2syl 16 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  B )
4 simpl3l 1012 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  D  e.  On )
5 eloni 4532 . . . . . . 7  |-  ( D  e.  On  ->  Ord  D )
64, 5syl 16 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  D )
7 ordtri3or 4554 . . . . . 6  |-  ( ( Ord  B  /\  Ord  D )  ->  ( B  e.  D  \/  B  =  D  \/  D  e.  B ) )
83, 6, 7syl2anc 643 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  \/  B  =  D  \/  D  e.  B ) )
9 simpr 448 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E ) )
10 simpl1l 1008 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  A  e.  On )
11 omcl 6716 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  D  e.  On )  ->  ( A  .o  D
)  e.  On )
1210, 4, 11syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( A  .o  D )  e.  On )
13 simpl3r 1013 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  E  e.  A
)
14 onelon 4547 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  E  e.  A )  ->  E  e.  On )
1510, 13, 14syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  E  e.  On )
16 oacl 6715 . . . . . . . . . . 11  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( ( A  .o  D )  +o  E
)  e.  On )
1712, 15, 16syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( A  .o  D )  +o  E )  e.  On )
18 eloni 4532 . . . . . . . . . 10  |-  ( ( ( A  .o  D
)  +o  E )  e.  On  ->  Ord  ( ( A  .o  D )  +o  E
) )
19 ordirr 4540 . . . . . . . . . 10  |-  ( Ord  ( ( A  .o  D )  +o  E
)  ->  -.  (
( A  .o  D
)  +o  E )  e.  ( ( A  .o  D )  +o  E ) )
2017, 18, 193syl 19 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  D
)  +o  E ) )
219, 20eqneltrd 2480 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) )
22 orc 375 . . . . . . . . 9  |-  ( B  e.  D  ->  ( B  e.  D  \/  ( B  =  D  /\  C  e.  E
) ) )
23 omeulem2 6762 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E ) )  -> 
( ( A  .o  B )  +o  C
)  e.  ( ( A  .o  D )  +o  E ) ) )
2423adantr 452 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E )
)  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
2522, 24syl5 30 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
2621, 25mtod 170 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  B  e.  D )
2726pm2.21d 100 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  ->  B  =  D ) )
28 idd 22 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  =  D  ->  B  =  D ) )
2920, 9neleqtrrd 2483 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) )
30 orc 375 . . . . . . . . 9  |-  ( D  e.  B  ->  ( D  e.  B  \/  ( D  =  B  /\  E  e.  C
) ) )
31 simpl1r 1009 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  A  =/=  (/) )
32 simpl2r 1011 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  C  e.  A
)
33 omeulem2 6762 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( D  e.  On  /\  E  e.  A )  /\  ( B  e.  On  /\  C  e.  A ) )  -> 
( ( D  e.  B  \/  ( D  =  B  /\  E  e.  C ) )  -> 
( ( A  .o  D )  +o  E
)  e.  ( ( A  .o  B )  +o  C ) ) )
3410, 31, 4, 13, 1, 32, 33syl222anc 1200 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( D  e.  B  \/  ( D  =  B  /\  E  e.  C )
)  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
3530, 34syl5 30 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( D  e.  B  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
3629, 35mtod 170 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  D  e.  B )
3736pm2.21d 100 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( D  e.  B  ->  B  =  D ) )
3827, 28, 373jaod 1248 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  e.  D  \/  B  =  D  \/  D  e.  B )  ->  B  =  D ) )
398, 38mpd 15 . . . 4  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  B  =  D )
40 onelon 4547 . . . . . . . 8  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
41 eloni 4532 . . . . . . . 8  |-  ( C  e.  On  ->  Ord  C )
4240, 41syl 16 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  A )  ->  Ord  C )
4310, 32, 42syl2anc 643 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  C )
44 eloni 4532 . . . . . . . 8  |-  ( E  e.  On  ->  Ord  E )
4514, 44syl 16 . . . . . . 7  |-  ( ( A  e.  On  /\  E  e.  A )  ->  Ord  E )
4610, 13, 45syl2anc 643 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  E )
47 ordtri3or 4554 . . . . . 6  |-  ( ( Ord  C  /\  Ord  E )  ->  ( C  e.  E  \/  C  =  E  \/  E  e.  C ) )
4843, 46, 47syl2anc 643 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  \/  C  =  E  \/  E  e.  C ) )
49 olc 374 . . . . . . . . . 10  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( B  e.  D  \/  ( B  =  D  /\  C  e.  E
) ) )
5049, 24syl5 30 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  =  D  /\  C  e.  E )  ->  (
( A  .o  B
)  +o  C )  e.  ( ( A  .o  D )  +o  E ) ) )
5139, 50mpand 657 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
5221, 51mtod 170 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  C  e.  E )
5352pm2.21d 100 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  ->  C  =  E ) )
54 idd 22 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  =  E  ->  C  =  E ) )
5539eqcomd 2392 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  D  =  B )
56 olc 374 . . . . . . . . . 10  |-  ( ( D  =  B  /\  E  e.  C )  ->  ( D  e.  B  \/  ( D  =  B  /\  E  e.  C
) ) )
5756, 34syl5 30 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( D  =  B  /\  E  e.  C )  ->  (
( A  .o  D
)  +o  E )  e.  ( ( A  .o  B )  +o  C ) ) )
5855, 57mpand 657 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( E  e.  C  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
5929, 58mtod 170 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  E  e.  C )
6059pm2.21d 100 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( E  e.  C  ->  C  =  E ) )
6153, 54, 603jaod 1248 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( C  e.  E  \/  C  =  E  \/  E  e.  C )  ->  C  =  E ) )
6248, 61mpd 15 . . . 4  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  C  =  E )
6339, 62jca 519 . . 3  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  =  D  /\  C  =  E ) )
6463ex 424 . 2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  ->  ( B  =  D  /\  C  =  E ) ) )
65 oveq2 6028 . . 3  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
66 id 20 . . 3  |-  ( C  =  E  ->  C  =  E )
6765, 66oveqan12d 6039 . 2  |-  ( ( B  =  D  /\  C  =  E )  ->  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )
6864, 67impbid1 195 1  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  <-> 
( B  =  D  /\  C  =  E ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   (/)c0 3571   Ord word 4521   Oncon0 4522  (class class class)co 6020    +o coa 6657    .o comu 6658
This theorem is referenced by:  omeu  6764  dfac12lem2  7957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-recs 6569  df-rdg 6604  df-oadd 6664  df-omul 6665
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