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Theorem on0eqel 5526
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 3767 . . 3  |-  (/)  C_  A
2 0elon 5462 . . . 4  |-  (/)  e.  On
3 onsseleq 5450 . . . 4  |-  ( (
(/)  e.  On  /\  A  e.  On )  ->  ( (/)  C_  A  <->  ( (/)  e.  A  \/  (/)  =  A ) ) )
42, 3mpan 668 . . 3  |-  ( A  e.  On  ->  ( (/)  C_  A  <->  ( (/)  e.  A  \/  (/)  =  A ) ) )
51, 4mpbii 211 . 2  |-  ( A  e.  On  ->  ( (/) 
e.  A  \/  (/)  =  A ) )
6 eqcom 2411 . . . 4  |-  ( (/)  =  A  <->  A  =  (/) )
76orbi2i 517 . . 3  |-  ( (
(/)  e.  A  \/  (/)  =  A )  <->  ( (/)  e.  A  \/  A  =  (/) ) )
8 orcom 385 . . 3  |-  ( (
(/)  e.  A  \/  A  =  (/) )  <->  ( A  =  (/)  \/  (/)  e.  A
) )
97, 8bitri 249 . 2  |-  ( (
(/)  e.  A  \/  (/)  =  A )  <->  ( A  =  (/)  \/  (/)  e.  A
) )
105, 9sylib 196 1  |-  ( A  e.  On  ->  ( A  =  (/)  \/  (/)  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    = wceq 1405    e. wcel 1842    C_ wss 3413   (/)c0 3737   Oncon0 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 5412  df-on 5413
This theorem is referenced by:  snsn0non  5527  onxpdisj  5528  omabs  7332  cnfcom3lem  8178  cnfcom3lemOLD  8186
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