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Theorem ordpss 40650
Description: ordelpss 6216 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordpss (Ord 𝐵 → (𝐴𝐵𝐴𝐵))

Proof of Theorem ordpss
StepHypRef Expression
1 ordelord 6210 . . . 4 ((Ord 𝐵𝐴𝐵) → Ord 𝐴)
21ex 413 . . 3 (Ord 𝐵 → (𝐴𝐵 → Ord 𝐴))
32ancrd 552 . 2 (Ord 𝐵 → (𝐴𝐵 → (Ord 𝐴𝐴𝐵)))
4 ordelpss 6216 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
54ancoms 459 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵𝐴𝐵))
65biimpd 230 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵𝐴𝐵))
76expimpd 454 . 2 (Ord 𝐵 → ((Ord 𝐴𝐴𝐵) → 𝐴𝐵))
83, 7syld 47 1 (Ord 𝐵 → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2107  wpss 3940  Ord word 6187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-ord 6191
This theorem is referenced by: (None)
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