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Theorem soss 4310
Description: Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
soss (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))

Proof of Theorem soss
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poss 4294 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
2 ssel 3149 . . . . . . . 8 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 ssel 3149 . . . . . . . 8 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
4 ssel 3149 . . . . . . . 8 (𝐴𝐵 → (𝑧𝐴𝑧𝐵))
52, 3, 43anim123d 1319 . . . . . . 7 (𝐴𝐵 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝐵𝑦𝐵𝑧𝐵)))
65imim1d 75 . . . . . 6 (𝐴𝐵 → (((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
762alimdv 1881 . . . . 5 (𝐴𝐵 → (∀𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ∀𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
87alimdv 1879 . . . 4 (𝐴𝐵 → (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
9 r3al 2521 . . . 4 (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
10 r3al 2521 . . . 4 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
118, 9, 103imtr4g 205 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
121, 11anim12d 335 . 2 (𝐴𝐵 → ((𝑅 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
13 df-iso 4293 . 2 (𝑅 Or 𝐵 ↔ (𝑅 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
14 df-iso 4293 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
1512, 13, 143imtr4g 205 1 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708  w3a 978  wal 1351  wcel 2148  wral 2455  wss 3129   class class class wbr 4000   Po wpo 4290   Or wor 4291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-in 3135  df-ss 3142  df-po 4292  df-iso 4293
This theorem is referenced by:  soeq2  4312
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