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Theorem onelpss 4555
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 4525 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4525 . 2  |-  ( B  e.  On  ->  Ord  B )
3 ordelssne 4542 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )
41, 2, 3syl2an 464 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717    =/= wne 2543    C_ wss 3256   Ord word 4514   Oncon0 4515
This theorem is referenced by:  tfindsg  4773  findsg  4805  oancom  7532  cardsdom2  7801  alephord  7882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519
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