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Theorem swoso 8172
 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 5372 . . 3 (𝜑< Po 𝑋)
5 poss 5364 . . 3 (𝑌𝑋 → ( < Po 𝑋< Po 𝑌))
61, 4, 5sylc 65 . 2 (𝜑< Po 𝑌)
71sselda 3889 . . . . . . 7 ((𝜑𝑥𝑌) → 𝑥𝑋)
81sselda 3889 . . . . . . 7 ((𝜑𝑦𝑌) → 𝑦𝑋)
97, 8anim12dan 618 . . . . . 6 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑋𝑦𝑋))
10 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
1110brdifun 8168 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦𝑦 < 𝑥)))
129, 11syl 17 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦𝑦 < 𝑥)))
13 df-3an 1082 . . . . . . 7 ((𝑥𝑌𝑦𝑌𝑥𝑅𝑦) ↔ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦))
14 swoso.5 . . . . . . 7 ((𝜑 ∧ (𝑥𝑌𝑦𝑌𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1513, 14sylan2br 594 . . . . . 6 ((𝜑 ∧ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1615expr 457 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦𝑥 = 𝑦))
1712, 16sylbird 261 . . . 4 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (¬ (𝑥 < 𝑦𝑦 < 𝑥) → 𝑥 = 𝑦))
1817orrd 858 . . 3 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
19 3orcomb 1087 . . . 4 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ (𝑥 < 𝑦𝑦 < 𝑥𝑥 = 𝑦))
20 df-3or 1081 . . . 4 ((𝑥 < 𝑦𝑦 < 𝑥𝑥 = 𝑦) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
2119, 20bitri 276 . . 3 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
2218, 21sylibr 235 . 2 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
236, 22issod 5394 1 (𝜑< Or 𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∨ w3o 1079   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ∖ cdif 3856   ∪ cun 3857   ⊆ wss 3859   class class class wbr 4962   Po wpo 5360   Or wor 5361   × cxp 5441  ◡ccnv 5442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-po 5362  df-so 5363  df-xp 5449  df-cnv 5451 This theorem is referenced by: (None)
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