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Theorem oa00 6998
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 4774 . . . . . . 7  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
21adantr 465 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
3 oaword1 6991 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
43sseld 3355 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  -> 
(/)  e.  ( A  +o  B ) ) )
52, 4sylbird 235 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  (/)  e.  ( A  +o  B ) ) )
6 ne0i 3643 . . . . 5  |-  ( (/)  e.  ( A  +o  B
)  ->  ( A  +o  B )  =/=  (/) )
75, 6syl6 33 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  ( A  +o  B )  =/=  (/) ) )
87necon4d 2674 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  ->  A  =  (/) ) )
9 on0eln0 4774 . . . . . . 7  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
109adantl 466 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  B  =/=  (/) ) )
11 0elon 4772 . . . . . . . 8  |-  (/)  e.  On
12 oaord 6986 . . . . . . . 8  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/) 
e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B ) ) )
1311, 12mp3an1 1301 . . . . . . 7  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B
) ) )
1413ancoms 453 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B
) ) )
1510, 14bitr3d 255 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =/=  (/)  <->  ( A  +o  (/) )  e.  ( A  +o  B ) ) )
16 ne0i 3643 . . . . 5  |-  ( ( A  +o  (/) )  e.  ( A  +o  B
)  ->  ( A  +o  B )  =/=  (/) )
1715, 16syl6bi 228 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  =/=  (/)  ->  ( A  +o  B )  =/=  (/) ) )
1817necon4d 2674 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  ->  B  =  (/) ) )
198, 18jcad 533 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) ) )
20 oveq12 6100 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  +o  B )  =  ( (/)  +o  (/) ) )
21 oa0 6956 . . . 4  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
2211, 21ax-mp 5 . . 3  |-  ( (/)  +o  (/) )  =  (/)
2320, 22syl6eq 2491 . 2  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  +o  B )  =  (/) )
2419, 23impbid1 203 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  <->  ( A  =  (/)  /\  B  =  (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   (/)c0 3637   Oncon0 4719  (class class class)co 6091    +o coa 6917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-oadd 6924
This theorem is referenced by:  oalimcl  6999  oeoa  7036
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