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Theorem on0eqel 4947
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 3777 . . 3  |-  (/)  C_  A
2 0elon 4883 . . . 4  |-  (/)  e.  On
3 onsseleq 4871 . . . 4  |-  ( (
(/)  e.  On  /\  A  e.  On )  ->  ( (/)  C_  A  <->  ( (/)  e.  A  \/  (/)  =  A ) ) )
42, 3mpan 670 . . 3  |-  ( A  e.  On  ->  ( (/)  C_  A  <->  ( (/)  e.  A  \/  (/)  =  A ) ) )
51, 4mpbii 211 . 2  |-  ( A  e.  On  ->  ( (/) 
e.  A  \/  (/)  =  A ) )
6 eqcom 2463 . . . 4  |-  ( (/)  =  A  <->  A  =  (/) )
76orbi2i 519 . . 3  |-  ( (
(/)  e.  A  \/  (/)  =  A )  <->  ( (/)  e.  A  \/  A  =  (/) ) )
8 orcom 387 . . 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 368    = wceq 1370    e. wcel 1758    C_ wss 3439   (/)c0 3748   Oncon0 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834
This theorem is referenced by:  snsn0non  4948  onxpdisj  5031  omabs  7199  cnfcom3lem  8051  cnfcom3lemOLD  8059
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