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Theorem ordelpss 4915
Description: Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
ordelpss  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )

Proof of Theorem ordelpss
StepHypRef Expression
1 ordelssne 4914 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )
2 df-pss 3487 . 2  |-  ( A 
C.  B  <->  ( A  C_  B  /\  A  =/= 
B ) )
31, 2syl6bbr 263 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1819    =/= wne 2652    C_ wss 3471    C. wpss 3472   Ord word 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890
This theorem is referenced by:  ordsseleq  4916  ordtri3or  4919  ordtr2  4931  onpsssuc  6653  php4  7723  nndomo  7730  omsucdomOLD  7732  ordpss  31564
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