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Theorem poss 5593
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 4051 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2 ss2ralv 4053 . . . 4 (𝐴𝐵 → (∀𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
32ralimdv 3168 . . 3 (𝐴𝐵 → (∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
41, 3syld 47 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
5 df-po 5591 . 2 (𝑅 Po 𝐵 ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
6 df-po 5591 . 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 3060  wss 3950   class class class wbr 5142   Po wpo 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909
This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3061  df-ss 3967  df-po 5591
This theorem is referenced by:  poeq2  5595  soss  5611  frpomin  6360  fprlem1  8326  swoso  8780  frfi  9322  wemapsolem  9591  fin23lem27  10369  zorn2lem6  10542  chnso  33005  xrge0iifiso  33935  incsequz2  37757
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