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Theorem swoord2 8779
Description: The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-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)))
swoord.4 (𝜑𝐵𝑋)
swoord.5 (𝜑𝐶𝑋)
swoord.6 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
swoord2 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
Distinct variable groups:   𝑥,𝑦,𝑧, <   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)

Proof of Theorem swoord2
StepHypRef Expression
1 id 22 . . . 4 (𝜑𝜑)
2 swoord.5 . . . 4 (𝜑𝐶𝑋)
3 swoord.6 . . . . 5 (𝜑𝐴𝑅𝐵)
4 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
5 difss 4135 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
64, 5eqsstri 4029 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
76ssbri 5187 . . . . 5 (𝐴𝑅𝐵𝐴(𝑋 × 𝑋)𝐵)
8 df-br 5143 . . . . . 6 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
9 opelxp1 5726 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴𝑋)
108, 9sylbi 217 . . . . 5 (𝐴(𝑋 × 𝑋)𝐵𝐴𝑋)
113, 7, 103syl 18 . . . 4 (𝜑𝐴𝑋)
12 swoord.4 . . . 4 (𝜑𝐵𝑋)
13 swoer.3 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1413swopolem 5601 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
151, 2, 11, 12, 14syl13anc 1373 . . 3 (𝜑 → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
16 idd 24 . . . 4 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐵))
174brdifun 8776 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
1811, 12, 17syl2anc 584 . . . . . . 7 (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
193, 18mpbid 232 . . . . . 6 (𝜑 → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
20 olc 868 . . . . . 6 (𝐵 < 𝐴 → (𝐴 < 𝐵𝐵 < 𝐴))
2119, 20nsyl 140 . . . . 5 (𝜑 → ¬ 𝐵 < 𝐴)
2221pm2.21d 121 . . . 4 (𝜑 → (𝐵 < 𝐴𝐶 < 𝐵))
2316, 22jaod 859 . . 3 (𝜑 → ((𝐶 < 𝐵𝐵 < 𝐴) → 𝐶 < 𝐵))
2415, 23syld 47 . 2 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
2513swopolem 5601 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐵𝑋𝐴𝑋)) → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
261, 2, 12, 11, 25syl13anc 1373 . . 3 (𝜑 → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
27 idd 24 . . . 4 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐴))
28 orc 867 . . . . . 6 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
2919, 28nsyl 140 . . . . 5 (𝜑 → ¬ 𝐴 < 𝐵)
3029pm2.21d 121 . . . 4 (𝜑 → (𝐴 < 𝐵𝐶 < 𝐴))
3127, 30jaod 859 . . 3 (𝜑 → ((𝐶 < 𝐴𝐴 < 𝐵) → 𝐶 < 𝐴))
3226, 31syld 47 . 2 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐴))
3324, 32impbid 212 1 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  cdif 3947  cun 3948  cop 4631   class class class wbr 5142   × cxp 5682  ccnv 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692
This theorem is referenced by: (None)
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