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Theorem poss 4961
 Description: Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
poss (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))

Proof of Theorem poss
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssralv 3629 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2 ssralv 3629 . . . . 5 (𝐴𝐵 → (∀𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
3 ssralv 3629 . . . . . 6 (𝐴𝐵 → (∀𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
43ralimdv 2946 . . . . 5 (𝐴𝐵 → (∀𝑦𝐴𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
52, 4syld 46 . . . 4 (𝐴𝐵 → (∀𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
65ralimdv 2946 . . 3 (𝐴𝐵 → (∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
71, 6syld 46 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
8 df-po 4959 . 2 (𝑅 Po 𝐵 ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
9 df-po 4959 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
107, 8, 93imtr4g 284 1 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wral 2896   ⊆ wss 3540   class class class wbr 4583   Po wpo 4957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-in 3547  df-ss 3554  df-po 4959 This theorem is referenced by:  poeq2  4963  soss  4977  swoso  7662  frfi  8090  wemapsolem  8338  fin23lem27  9033  zorn2lem6  9206  xrge0iifiso  29309  incsequz2  32715
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