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Theorem onelpss 4860
Description: Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
onelpss  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B ) ) )

Proof of Theorem onelpss
StepHypRef Expression
1 eloni 4830 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4830 . 2  |-  ( B  e.  On  ->  Ord  B )
3 ordelssne 4847 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )
41, 2, 3syl2an 477 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    =/= wne 2644    C_ wss 3429   Ord word 4819   Oncon0 4820
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824
This theorem is referenced by:  tfindsg  6574  findsg  6606  oancom  7961  cardsdom2  8262  alephord  8349
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