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Theorem ordpss 39423
Description: ordelpss 5967 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 5961 . . . 4 ((Ord 𝐵𝐴𝐵) → Ord 𝐴)
21ex 402 . . 3 (Ord 𝐵 → (𝐴𝐵 → Ord 𝐴))
32ancrd 548 . 2 (Ord 𝐵 → (𝐴𝐵 → (Ord 𝐴𝐴𝐵)))
4 ordelpss 5967 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
54ancoms 451 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵𝐴𝐵))
65biimpd 221 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵𝐴𝐵))
76expimpd 446 . 2 (Ord 𝐵 → ((Ord 𝐴𝐴𝐵) → 𝐴𝐵))
83, 7syld 47 1 (Ord 𝐵 → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wpss 3768  Ord word 5938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-tr 4944  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-ord 5942
This theorem is referenced by: (None)
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