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Theorem ordsseleq 4897
Description: For ordinal classes, subclass is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordsseleq  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )

Proof of Theorem ordsseleq
StepHypRef Expression
1 ordelpss 4896 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
21orbi1d 702 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  <->  ( A  C.  B  \/  A  =  B )
) )
3 sspss 3588 . 2  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
42, 3syl6rbbr 264 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804    C_ wss 3461    C. wpss 3462   Ord word 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871
This theorem is referenced by:  ordtri3or  4900  ordtri1  4901  ordtri2  4903  onsseleq  4909  ordsssuc  4954  ordsson  6610  ordsucelsuc  6642  limom  6700  onfununi  7014  cfslbn  8650
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