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Theorem swoord1 7660
 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
swoord1 (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))
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 swoord1
StepHypRef Expression
1 id 22 . . . 4 (𝜑𝜑)
2 swoord.6 . . . . 5 (𝜑𝐴𝑅𝐵)
3 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
4 difss 3699 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
53, 4eqsstri 3598 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
65ssbri 4627 . . . . 5 (𝐴𝑅𝐵𝐴(𝑋 × 𝑋)𝐵)
7 df-br 4584 . . . . . 6 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
8 opelxp1 5074 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴𝑋)
97, 8sylbi 206 . . . . 5 (𝐴(𝑋 × 𝑋)𝐵𝐴𝑋)
102, 6, 93syl 18 . . . 4 (𝜑𝐴𝑋)
11 swoord.5 . . . 4 (𝜑𝐶𝑋)
12 swoord.4 . . . 4 (𝜑𝐵𝑋)
13 swoer.3 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1413swopolem 4968 . . . 4 ((𝜑 ∧ (𝐴𝑋𝐶𝑋𝐵𝑋)) → (𝐴 < 𝐶 → (𝐴 < 𝐵𝐵 < 𝐶)))
151, 10, 11, 12, 14syl13anc 1320 . . 3 (𝜑 → (𝐴 < 𝐶 → (𝐴 < 𝐵𝐵 < 𝐶)))
163brdifun 7658 . . . . . . 7 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
1710, 12, 16syl2anc 691 . . . . . 6 (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
182, 17mpbid 221 . . . . 5 (𝜑 → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
19 orc 399 . . . . 5 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
2018, 19nsyl 134 . . . 4 (𝜑 → ¬ 𝐴 < 𝐵)
21 biorf 419 . . . 4 𝐴 < 𝐵 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
2220, 21syl 17 . . 3 (𝜑 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
2315, 22sylibrd 248 . 2 (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))
2413swopolem 4968 . . . 4 ((𝜑 ∧ (𝐵𝑋𝐶𝑋𝐴𝑋)) → (𝐵 < 𝐶 → (𝐵 < 𝐴𝐴 < 𝐶)))
251, 12, 11, 10, 24syl13anc 1320 . . 3 (𝜑 → (𝐵 < 𝐶 → (𝐵 < 𝐴𝐴 < 𝐶)))
26 olc 398 . . . . 5 (𝐵 < 𝐴 → (𝐴 < 𝐵𝐵 < 𝐴))
2718, 26nsyl 134 . . . 4 (𝜑 → ¬ 𝐵 < 𝐴)
28 biorf 419 . . . 4 𝐵 < 𝐴 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴𝐴 < 𝐶)))
2927, 28syl 17 . . 3 (𝜑 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴𝐴 < 𝐶)))
3025, 29sylibrd 248 . 2 (𝜑 → (𝐵 < 𝐶𝐴 < 𝐶))
3123, 30impbid 201 1 (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ◡ccnv 5037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046 This theorem is referenced by: (None)
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