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Theorem soss 4115
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 4099 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
2 ssel 3008 . . . . . . . 8 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 ssel 3008 . . . . . . . 8 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
4 ssel 3008 . . . . . . . 8 (𝐴𝐵 → (𝑧𝐴𝑧𝐵))
52, 3, 43anim123d 1253 . . . . . . 7 (𝐴𝐵 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝐵𝑦𝐵𝑧𝐵)))
65imim1d 74 . . . . . 6 (𝐴𝐵 → (((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
762alimdv 1806 . . . . 5 (𝐴𝐵 → (∀𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ∀𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
87alimdv 1804 . . . 4 (𝐴𝐵 → (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
9 r3al 2416 . . . 4 (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
10 r3al 2416 . . . 4 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
118, 9, 103imtr4g 203 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
121, 11anim12d 328 . 2 (𝐴𝐵 → ((𝑅 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
13 df-iso 4098 . 2 (𝑅 Or 𝐵 ↔ (𝑅 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
14 df-iso 4098 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
1512, 13, 143imtr4g 203 1 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662  w3a 922  wal 1285  wcel 1436  wral 2355  wss 2988   class class class wbr 3820   Po wpo 4095   Or wor 4096
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-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  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-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-in 2994  df-ss 3001  df-po 4097  df-iso 4098
This theorem is referenced by:  soeq2  4117
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