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Theorem onelpss 4448
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 4418 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4418 . 2  |-  ( B  e.  On  ->  Ord  B )
3 ordelssne 4435 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )
41, 2, 3syl2an 463 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 176    /\ wa 358    e. wcel 1696    =/= wne 2459    C_ wss 3165   Ord word 4407   Oncon0 4408
This theorem is referenced by:  tfindsg  4667  findsg  4699  oancom  7368  cardsdom2  7637  alephord  7718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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