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Theorem soss 4079
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 4063 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
2 ssel 2967 . . . . . . . 8 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 ssel 2967 . . . . . . . 8 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
4 ssel 2967 . . . . . . . 8 (𝐴𝐵 → (𝑧𝐴𝑧𝐵))
52, 3, 43anim123d 1225 . . . . . . 7 (𝐴𝐵 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝐵𝑦𝐵𝑧𝐵)))
65imim1d 73 . . . . . 6 (𝐴𝐵 → (((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
762alimdv 1777 . . . . 5 (𝐴𝐵 → (∀𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ∀𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
87alimdv 1775 . . . 4 (𝐴𝐵 → (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
9 r3al 2383 . . . 4 (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
10 r3al 2383 . . . 4 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
118, 9, 103imtr4g 198 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
121, 11anim12d 322 . 2 (𝐴𝐵 → ((𝑅 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
13 df-iso 4062 . 2 (𝑅 Or 𝐵 ↔ (𝑅 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
14 df-iso 4062 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
1512, 13, 143imtr4g 198 1 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 639  w3a 896  wal 1257  wcel 1409  wral 2323  wss 2945   class class class wbr 3792   Po wpo 4059   Or wor 4060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-in 2952  df-ss 2959  df-po 4061  df-iso 4062
This theorem is referenced by:  soeq2  4081
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