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Theorem on0eqel 5559
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 3793 . . 3  |-  (/)  C_  A
2 0elon 5495 . . . 4  |-  (/)  e.  On
3 onsseleq 5483 . . . 4  |-  ( (
(/)  e.  On  /\  A  e.  On )  ->  ( (/)  C_  A  <->  ( (/)  e.  A  \/  (/)  =  A ) ) )
42, 3mpan 674 . . 3  |-  ( A  e.  On  ->  ( (/)  C_  A  <->  ( (/)  e.  A  \/  (/)  =  A ) ) )
51, 4mpbii 214 . 2  |-  ( A  e.  On  ->  ( (/) 
e.  A  \/  (/)  =  A ) )
6 eqcom 2431 . . . 4  |-  ( (/)  =  A  <->  A  =  (/) )
76orbi2i 521 . . 3  |-  ( (
(/)  e.  A  \/  (/)  =  A )  <->  ( (/)  e.  A  \/  A  =  (/) ) )
8 orcom 388 . . 3  |-  ( (
(/)  e.  A  \/  A  =  (/) )  <->  ( A  =  (/)  \/  (/)  e.  A
) )
97, 8bitri 252 . 2  |-  ( (
(/)  e.  A  \/  (/)  =  A )  <->  ( A  =  (/)  \/  (/)  e.  A
) )
105, 9sylib 199 1  |-  ( A  e.  On  ->  ( A  =  (/)  \/  (/)  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    = wceq 1437    e. wcel 1872    C_ wss 3436   (/)c0 3761   Oncon0 5442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446
This theorem is referenced by:  snsn0non  5560  onxpdisj  5561  omabs  7359  cnfcom3lem  8216
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