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Theorem oa00 7220
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 4939 . . . . . . 7  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
21adantr 465 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
3 oaword1 7213 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
43sseld 3508 . . . . . 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 3796 . . . . 5  |-  ( (/)  e.  ( A  +o  B
)  ->  ( A  +o  B )  =/=  (/) )
75, 6syl6 33 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =/=  (/)  ->  ( A  +o  B )  =/=  (/) ) )
87necon4d 2694 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/)  ->  A  =  (/) ) )
9 on0eln0 4939 . . . . . . 7  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
109adantl 466 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  B  =/=  (/) ) )
11 0elon 4937 . . . . . . . 8  |-  (/)  e.  On
12 oaord 7208 . . . . . . . 8  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/) 
e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B ) ) )
1311, 12mp3an1 1311 . . . . . . 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 3796 . . . . 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 2694 . . 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 6304 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  +o  B )  =  ( (/)  +o  (/) ) )
21 oa0 7178 . . . 4  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
2211, 21ax-mp 5 . . 3  |-  ( (/)  +o  (/) )  =  (/)
2320, 22syl6eq 2524 . 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 1379    e. wcel 1767    =/= wne 2662   (/)c0 3790   Oncon0 4884  (class class class)co 6295    +o coa 7139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-oadd 7146
This theorem is referenced by:  oalimcl  7221  oeoa  7258
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