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Theorem poss 5173
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 3815 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2 ssralv 3815 . . . . 5 (𝐴𝐵 → (∀𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
3 ssralv 3815 . . . . . 6 (𝐴𝐵 → (∀𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
43ralimdv 3112 . . . . 5 (𝐴𝐵 → (∀𝑦𝐴𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
52, 4syld 47 . . . 4 (𝐴𝐵 → (∀𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
65ralimdv 3112 . . 3 (𝐴𝐵 → (∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
71, 6syld 47 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
8 df-po 5171 . 2 (𝑅 Po 𝐵 ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
9 df-po 5171 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
107, 8, 93imtr4g 285 1 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wral 3061  wss 3723   class class class wbr 4787   Po wpo 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ral 3066  df-in 3730  df-ss 3737  df-po 5171
This theorem is referenced by:  poeq2  5175  soss  5189  swoso  7933  frfi  8365  wemapsolem  8615  fin23lem27  9356  zorn2lem6  9529  xrge0iifiso  30321  frpomin  32075  incsequz2  33875
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