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Theorem ordpss 44446
Description: ordelpss 6413 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 6407 . . . 4 ((Ord 𝐵𝐴𝐵) → Ord 𝐴)
21ex 412 . . 3 (Ord 𝐵 → (𝐴𝐵 → Ord 𝐴))
32ancrd 551 . 2 (Ord 𝐵 → (𝐴𝐵 → (Ord 𝐴𝐴𝐵)))
4 ordelpss 6413 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
54ancoms 458 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵𝐴𝐵))
65biimpd 229 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵𝐴𝐵))
76expimpd 453 . 2 (Ord 𝐵 → ((Ord 𝐴𝐴𝐵) → 𝐴𝐵))
83, 7syld 47 1 (Ord 𝐵 → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2105  wpss 3963  Ord word 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388
This theorem is referenced by: (None)
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