MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swoso Structured version   Visualization version   GIF version

Theorem swoso 7727
Description: If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
swoer.1 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
swoer.2 ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
swoer.3 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
swoso.4 (𝜑𝑌𝑋)
swoso.5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌𝑥𝑅𝑦)) → 𝑥 = 𝑦)
Assertion
Ref Expression
swoso (𝜑< Or 𝑌)
Distinct variable groups:   𝑥,𝑦,𝑧, <   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)   𝑌(𝑧)

Proof of Theorem swoso
StepHypRef Expression
1 swoso.4 . . 3 (𝜑𝑌𝑋)
2 swoer.2 . . . 4 ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
3 swoer.3 . . . 4 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
42, 3swopo 5010 . . 3 (𝜑< Po 𝑋)
5 poss 5002 . . 3 (𝑌𝑋 → ( < Po 𝑋< Po 𝑌))
61, 4, 5sylc 65 . 2 (𝜑< Po 𝑌)
71sselda 3587 . . . . . . 7 ((𝜑𝑥𝑌) → 𝑥𝑋)
81sselda 3587 . . . . . . 7 ((𝜑𝑦𝑌) → 𝑦𝑋)
97, 8anim12dan 881 . . . . . 6 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑋𝑦𝑋))
10 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
1110brdifun 7723 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦𝑦 < 𝑥)))
129, 11syl 17 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦𝑦 < 𝑥)))
13 df-3an 1038 . . . . . . 7 ((𝑥𝑌𝑦𝑌𝑥𝑅𝑦) ↔ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦))
14 swoso.5 . . . . . . 7 ((𝜑 ∧ (𝑥𝑌𝑦𝑌𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1513, 14sylan2br 493 . . . . . 6 ((𝜑 ∧ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1615expr 642 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦𝑥 = 𝑦))
1712, 16sylbird 250 . . . 4 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (¬ (𝑥 < 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
1817orrd 393 . . 3 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
19 3orcomb 1046 . . . 4 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ (𝑥 < 𝑦𝑦 < 𝑥𝑥 = 𝑦))
20 df-3or 1037 . . . 4 ((𝑥 < 𝑦𝑦 < 𝑥𝑥 = 𝑦) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
2119, 20bitri 264 . . 3 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
2218, 21sylibr 224 . 2 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
236, 22issod 5030 1 (𝜑< Or 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  cdif 3556  cun 3557  wss 3559   class class class wbr 4618   Po wpo 4998   Or wor 4999   × cxp 5077  ccnv 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-po 5000  df-so 5001  df-xp 5085  df-cnv 5087
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator