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Theorem on0eqel 4510
Description: An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
Assertion
Ref Expression
on0eqel  |-  ( A  e.  On  ->  ( A  =  (/)  \/  (/)  e.  A
) )

Proof of Theorem on0eqel
StepHypRef Expression
1 0ss 3483 . . 3  |-  (/)  C_  A
2 0elon 4445 . . . 4  |-  (/)  e.  On
3 onsseleq 4433 . . . 4  |-  ( (
(/)  e.  On  /\  A  e.  On )  ->  ( (/)  C_  A  <->  ( (/)  e.  A  \/  (/)  =  A ) ) )
42, 3mpan 651 . . 3  |-  ( A  e.  On  ->  ( (/)  C_  A  <->  ( (/)  e.  A  \/  (/)  =  A ) ) )
51, 4mpbii 202 . 2  |-  ( A  e.  On  ->  ( (/) 
e.  A  \/  (/)  =  A ) )
6 eqcom 2285 . . . 4  |-  ( (/)  =  A  <->  A  =  (/) )
76orbi2i 505 . . 3  |-  ( (
(/)  e.  A  \/  (/)  =  A )  <->  ( (/)  e.  A  \/  A  =  (/) ) )
8 orcom 376 . . 3  |-  ( (
(/)  e.  A  \/  A  =  (/) )  <->  ( A  =  (/)  \/  (/)  e.  A
) )
97, 8bitri 240 . 2  |-  ( (
(/)  e.  A  \/  (/)  =  A )  <->  ( A  =  (/)  \/  (/)  e.  A
) )
105, 9sylib 188 1  |-  ( A  e.  On  ->  ( A  =  (/)  \/  (/)  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   Oncon0 4392
This theorem is referenced by:  snsn0non  4511  onxpdisj  4769  omabs  6645  cnfcom3lem  7406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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