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Theorem ordpss 27643
Description: ordelpss 4612 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordpss  |-  ( Ord 
B  ->  ( A  e.  B  ->  A  C.  B ) )

Proof of Theorem ordpss
StepHypRef Expression
1 ordelord 4606 . . . 4  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
21ex 425 . . 3  |-  ( Ord 
B  ->  ( A  e.  B  ->  Ord  A
) )
32ancrd 539 . 2  |-  ( Ord 
B  ->  ( A  e.  B  ->  ( Ord 
A  /\  A  e.  B ) ) )
4 ordelpss 4612 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
54ancoms 441 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  A  C.  B ) )
65biimpd 200 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  ->  A  C.  B ) )
76expimpd 588 . 2  |-  ( Ord 
B  ->  ( ( Ord  A  /\  A  e.  B )  ->  A  C.  B ) )
83, 7syld 43 1  |-  ( Ord 
B  ->  ( A  e.  B  ->  A  C.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726    C. wpss 3323   Ord word 4583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587
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