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Theorem onacda 7970
Description: The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
onacda  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  ~~  ( A  +c  B ) )

Proof of Theorem onacda
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 enrefg 7036 . . . . 5  |-  ( A  e.  On  ->  A  ~~  A )
21adantr 451 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  ~~  A )
3 simpr 447 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  e.  On )
4 eqid 2366 . . . . . . . 8  |-  ( x  e.  B  |->  ( A  +o  x ) )  =  ( x  e.  B  |->  ( A  +o  x ) )
54oacomf1olem 6704 . . . . . . 7  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( ( x  e.  B  |->  ( A  +o  x ) ) : B -1-1-onto-> ran  ( x  e.  B  |->  ( A  +o  x ) )  /\  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i 
A )  =  (/) ) )
65ancoms 439 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  B  |->  ( A  +o  x ) ) : B -1-1-onto-> ran  ( x  e.  B  |->  ( A  +o  x ) )  /\  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i 
A )  =  (/) ) )
76simpld 445 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  B  |->  ( A  +o  x
) ) : B -1-1-onto-> ran  ( x  e.  B  |->  ( A  +o  x
) ) )
8 f1oeng 7023 . . . . 5  |-  ( ( B  e.  On  /\  ( x  e.  B  |->  ( A  +o  x
) ) : B -1-1-onto-> ran  ( x  e.  B  |->  ( A  +o  x
) ) )  ->  B  ~~  ran  ( x  e.  B  |->  ( A  +o  x ) ) )
93, 7, 8syl2anc 642 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  ~~  ran  (
x  e.  B  |->  ( A  +o  x ) ) )
10 incom 3449 . . . . 5  |-  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x ) ) )  =  ( ran  ( x  e.  B  |->  ( A  +o  x
) )  i^i  A
)
116simprd 449 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  ( x  e.  B  |->  ( A  +o  x ) )  i^i  A )  =  (/) )
1210, 11syl5eq 2410 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x
) ) )  =  (/) )
13 cdaenun 7947 . . . 4  |-  ( ( A  ~~  A  /\  B  ~~  ran  ( x  e.  B  |->  ( A  +o  x ) )  /\  ( A  i^i  ran  ( x  e.  B  |->  ( A  +o  x
) ) )  =  (/) )  ->  ( A  +c  B )  ~~  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x
) ) ) )
142, 9, 12, 13syl3anc 1183 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
15 oarec 6702 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( A  u.  ran  ( x  e.  B  |->  ( A  +o  x ) ) ) )
1614, 15breqtrrd 4151 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +c  B
)  ~~  ( A  +o  B ) )
17 ensym 7053 . 2  |-  ( ( A  +c  B ) 
~~  ( A  +o  B )  ->  ( A  +o  B )  ~~  ( A  +c  B
) )
1816, 17syl 15 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  ~~  ( A  +c  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    u. cun 3236    i^i cin 3237   (/)c0 3543   class class class wbr 4125    e. cmpt 4179   Oncon0 4495   ran crn 4793   -1-1-onto->wf1o 5357  (class class class)co 5981    +o coa 6618    ~~ cen 7003    +c ccda 7940
This theorem is referenced by:  cardacda  7971  nnacda  7974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-cda 7941
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