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Theorem swoso 7946
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 5197 . . 3 (𝜑< Po 𝑋)
5 poss 5189 . . 3 (𝑌𝑋 → ( < Po 𝑋< Po 𝑌))
61, 4, 5sylc 65 . 2 (𝜑< Po 𝑌)
71sselda 3744 . . . . . . 7 ((𝜑𝑥𝑌) → 𝑥𝑋)
81sselda 3744 . . . . . . 7 ((𝜑𝑦𝑌) → 𝑦𝑋)
97, 8anim12dan 918 . . . . . 6 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑋𝑦𝑋))
10 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
1110brdifun 7942 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦𝑦 < 𝑥)))
129, 11syl 17 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦𝑦 < 𝑥)))
13 df-3an 1074 . . . . . . 7 ((𝑥𝑌𝑦𝑌𝑥𝑅𝑦) ↔ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦))
14 swoso.5 . . . . . . 7 ((𝜑 ∧ (𝑥𝑌𝑦𝑌𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1513, 14sylan2br 494 . . . . . 6 ((𝜑 ∧ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1615expr 644 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦𝑥 = 𝑦))
1712, 16sylbird 250 . . . 4 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (¬ (𝑥 < 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
1817orrd 392 . . 3 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
19 3orcomb 1079 . . . 4 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ (𝑥 < 𝑦𝑦 < 𝑥𝑥 = 𝑦))
20 df-3or 1073 . . . 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 5217 1 (𝜑< Or 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3o 1071  w3a 1072   = wceq 1632  wcel 2139  cdif 3712  cun 3713  wss 3715   class class class wbr 4804   Po wpo 5185   Or wor 5186   × cxp 5264  ccnv 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-po 5187  df-so 5188  df-xp 5272  df-cnv 5274
This theorem is referenced by: (None)
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