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Theorem swoso 8332
 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 5453 . . 3 (𝜑< Po 𝑋)
5 poss 5445 . . 3 (𝑌𝑋 → ( < Po 𝑋< Po 𝑌))
61, 4, 5sylc 65 . 2 (𝜑< Po 𝑌)
71sselda 3892 . . . . . . 7 ((𝜑𝑥𝑌) → 𝑥𝑋)
81sselda 3892 . . . . . . 7 ((𝜑𝑦𝑌) → 𝑦𝑋)
97, 8anim12dan 621 . . . . . 6 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑋𝑦𝑋))
10 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
1110brdifun 8328 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦𝑦 < 𝑥)))
129, 11syl 17 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦𝑦 < 𝑥)))
13 df-3an 1086 . . . . . . 7 ((𝑥𝑌𝑦𝑌𝑥𝑅𝑦) ↔ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦))
14 swoso.5 . . . . . . 7 ((𝜑 ∧ (𝑥𝑌𝑦𝑌𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1513, 14sylan2br 597 . . . . . 6 ((𝜑 ∧ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1615expr 460 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦𝑥 = 𝑦))
1712, 16sylbird 263 . . . 4 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (¬ (𝑥 < 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
1817orrd 860 . . 3 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
19 3orcomb 1091 . . . 4 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ (𝑥 < 𝑦𝑦 < 𝑥𝑥 = 𝑦))
20 df-3or 1085 . . . 4 ((𝑥 < 𝑦𝑦 < 𝑥𝑥 = 𝑦) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
2119, 20bitri 278 . . 3 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
2218, 21sylibr 237 . 2 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
236, 22issod 5475 1 (𝜑< Or 𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∖ cdif 3855   ∪ cun 3856   ⊆ wss 3858   class class class wbr 5032   Po wpo 5441   Or wor 5442   × cxp 5522  ◡ccnv 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-po 5443  df-so 5444  df-xp 5530  df-cnv 5532 This theorem is referenced by: (None)
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