ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  soss GIF version

Theorem soss 4206
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 4190 . . 3 (𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))
2 ssel 3061 . . . . . . . 8 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 ssel 3061 . . . . . . . 8 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
4 ssel 3061 . . . . . . . 8 (𝐴𝐵 → (𝑧𝐴𝑧𝐵))
52, 3, 43anim123d 1282 . . . . . . 7 (𝐴𝐵 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝐵𝑦𝐵𝑧𝐵)))
65imim1d 75 . . . . . 6 (𝐴𝐵 → (((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
762alimdv 1837 . . . . 5 (𝐴𝐵 → (∀𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ∀𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
87alimdv 1835 . . . 4 (𝐴𝐵 → (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
9 r3al 2454 . . . 4 (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
10 r3al 2454 . . . 4 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
118, 9, 103imtr4g 204 . . 3 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
121, 11anim12d 333 . 2 (𝐴𝐵 → ((𝑅 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))))
13 df-iso 4189 . 2 (𝑅 Or 𝐵 ↔ (𝑅 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
14 df-iso 4189 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
1512, 13, 143imtr4g 204 1 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 682  w3a 947  wal 1314  wcel 1465  wral 2393  wss 3041   class class class wbr 3899   Po wpo 4186   Or wor 4187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-in 3047  df-ss 3054  df-po 4188  df-iso 4189
This theorem is referenced by:  soeq2  4208
  Copyright terms: Public domain W3C validator