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Theorem poss 5599
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 4064 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2 ss2ralv 4066 . . . 4 (𝐴𝐵 → (∀𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
32ralimdv 3167 . . 3 (𝐴𝐵 → (∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
41, 3syld 47 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
5 df-po 5597 . 2 (𝑅 Po 𝐵 ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
6 df-po 5597 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
74, 5, 63imtr4g 296 1 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wral 3059  wss 3963   class class class wbr 5148   Po wpo 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908
This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3060  df-ss 3980  df-po 5597
This theorem is referenced by:  poeq2  5601  soss  5617  frpomin  6363  fprlem1  8324  swoso  8778  frfi  9319  wemapsolem  9588  fin23lem27  10366  zorn2lem6  10539  chnso  32988  xrge0iifiso  33896  incsequz2  37736
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