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Theorem swoso 8733
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 5590 . . 3 (𝜑< Po 𝑋)
5 poss 5581 . . 3 (𝑌𝑋 → ( < Po 𝑋< Po 𝑌))
61, 4, 5sylc 65 . 2 (𝜑< Po 𝑌)
71sselda 3975 . . . . . . 7 ((𝜑𝑥𝑌) → 𝑥𝑋)
81sselda 3975 . . . . . . 7 ((𝜑𝑦𝑌) → 𝑦𝑋)
97, 8anim12dan 618 . . . . . 6 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑋𝑦𝑋))
10 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
1110brdifun 8729 . . . . . 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 594 . . . . . 6 ((𝜑 ∧ ((𝑥𝑌𝑦𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
1615expr 456 . . . . 5 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝑅𝑦𝑥 = 𝑦))
1712, 16sylbird 260 . . . 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 275 . . 3 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
2218, 21sylibr 233 . 2 ((𝜑 ∧ (𝑥𝑌𝑦𝑌)) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
236, 22issod 5612 1 (𝜑< Or 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844  w3o 1083  w3a 1084   = wceq 1533  wcel 2098  cdif 3938  cun 3939  wss 3941   class class class wbr 5139   Po wpo 5577   Or wor 5578   × cxp 5665  ccnv 5666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-po 5579  df-so 5580  df-xp 5673  df-cnv 5675
This theorem is referenced by: (None)
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