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Theorem poss 4098
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 3074 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2 ssralv 3074 . . . . 5 (𝐴𝐵 → (∀𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
3 ssralv 3074 . . . . . 6 (𝐴𝐵 → (∀𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
43ralimdv 2438 . . . . 5 (𝐴𝐵 → (∀𝑦𝐴𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
52, 4syld 44 . . . 4 (𝐴𝐵 → (∀𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
65ralimdv 2438 . . 3 (𝐴𝐵 → (∀𝑥𝐴𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
71, 6syld 44 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
8 df-po 4096 . 2 (𝑅 Po 𝐵 ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
9 df-po 4096 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
107, 8, 93imtr4g 203 1 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wral 2355  wss 2988   class class class wbr 3820   Po wpo 4094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-ral 2360  df-in 2994  df-ss 3001  df-po 4096
This theorem is referenced by:  poeq2  4100  soss  4114  fimaxq  10124
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