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Theorem ordpss 27529
Description: ordelpss 4577 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 4571 . . . 4  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
21ex 424 . . 3  |-  ( Ord 
B  ->  ( A  e.  B  ->  Ord  A
) )
32ancrd 538 . 2  |-  ( Ord 
B  ->  ( A  e.  B  ->  ( Ord 
A  /\  A  e.  B ) ) )
4 ordelpss 4577 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
54ancoms 440 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  A  C.  B ) )
65biimpd 199 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  ->  A  C.  B ) )
76expimpd 587 . 2  |-  ( Ord 
B  ->  ( ( Ord  A  /\  A  e.  B )  ->  A  C.  B ) )
83, 7syld 42 1  |-  ( Ord 
B  ->  ( A  e.  B  ->  A  C.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721    C. wpss 3289   Ord word 4548
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552
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