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Theorem ordpss 36172
Description: ordelpss 4847 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 4841 . . . 4  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
21ex 432 . . 3  |-  ( Ord 
B  ->  ( A  e.  B  ->  Ord  A
) )
32ancrd 552 . 2  |-  ( Ord 
B  ->  ( A  e.  B  ->  ( Ord 
A  /\  A  e.  B ) ) )
4 ordelpss 4847 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
54ancoms 451 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  A  C.  B ) )
65biimpd 207 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  ->  A  C.  B ) )
76expimpd 601 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1840    C. wpss 3412   Ord word 4818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-tr 4487  df-eprel 4731  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822
This theorem is referenced by: (None)
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