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Theorem oa00 6804
Description: An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.)
Assertion
Ref Expression
oa00  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  <->  ( A  =  (/)  /\  B  =  (/) ) ) )

Proof of Theorem oa00
StepHypRef Expression
1 on0eln0 4638 . . . . . . 7  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
21adantr 453 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
3 oaword1 6797 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
43sseld 3349 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  -> 
(/)  e.  ( A  +o  B ) ) )
52, 4sylbird 228 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  (/)  e.  ( A  +o  B ) ) )
6 ne0i 3636 . . . . 5  |-  ( (/)  e.  ( A  +o  B
)  ->  ( A  +o  B )  =/=  (/) )
75, 6syl6 32 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  ( A  +o  B )  =/=  (/) ) )
87necon4d 2669 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  ->  A  =  (/) ) )
9 on0eln0 4638 . . . . . . 7  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
109adantl 454 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  B  =/=  (/) ) )
11 0elon 4636 . . . . . . . 8  |-  (/)  e.  On
12 oaord 6792 . . . . . . . 8  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/) 
e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B ) ) )
1311, 12mp3an1 1267 . . . . . . 7  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B
) ) )
1413ancoms 441 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B
) ) )
1510, 14bitr3d 248 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =/=  (/)  <->  ( A  +o  (/) )  e.  ( A  +o  B ) ) )
16 ne0i 3636 . . . . 5  |-  ( ( A  +o  (/) )  e.  ( A  +o  B
)  ->  ( A  +o  B )  =/=  (/) )
1715, 16syl6bi 221 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =/=  (/)  ->  ( A  +o  B )  =/=  (/) ) )
1817necon4d 2669 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  ->  B  =  (/) ) )
198, 18jcad 521 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) ) )
20 oveq12 6092 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  +o  B )  =  ( (/)  +o  (/) ) )
21 oa0 6762 . . . 4  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
2211, 21ax-mp 8 . . 3  |-  ( (/)  +o  (/) )  =  (/)
2320, 22syl6eq 2486 . 2  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  +o  B )  =  (/) )
2419, 23impbid1 196 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  <->  ( A  =  (/)  /\  B  =  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   Oncon0 4583  (class class class)co 6083    +o coa 6723
This theorem is referenced by:  oalimcl  6805  oeoa  6842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-oadd 6730
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