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Theorem swoord1 7725
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 3720 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
53, 4eqsstri 3619 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
65ssbri 4662 . . . . 5 (𝐴𝑅𝐵𝐴(𝑋 × 𝑋)𝐵)
7 df-br 4619 . . . . . 6 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
8 opelxp1 5115 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴𝑋)
97, 8sylbi 207 . . . . 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 5009 . . . 4 ((𝜑 ∧ (𝐴𝑋𝐶𝑋𝐵𝑋)) → (𝐴 < 𝐶 → (𝐴 < 𝐵𝐵 < 𝐶)))
151, 10, 11, 12, 14syl13anc 1325 . . 3 (𝜑 → (𝐴 < 𝐶 → (𝐴 < 𝐵𝐵 < 𝐶)))
163brdifun 7723 . . . . . . 7 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
1710, 12, 16syl2anc 692 . . . . . 6 (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
182, 17mpbid 222 . . . . 5 (𝜑 → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
19 orc 400 . . . . 5 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
2018, 19nsyl 135 . . . 4 (𝜑 → ¬ 𝐴 < 𝐵)
21 biorf 420 . . . 4 𝐴 < 𝐵 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
2220, 21syl 17 . . 3 (𝜑 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
2315, 22sylibrd 249 . 2 (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))
2413swopolem 5009 . . . 4 ((𝜑 ∧ (𝐵𝑋𝐶𝑋𝐴𝑋)) → (𝐵 < 𝐶 → (𝐵 < 𝐴𝐴 < 𝐶)))
251, 12, 11, 10, 24syl13anc 1325 . . 3 (𝜑 → (𝐵 < 𝐶 → (𝐵 < 𝐴𝐴 < 𝐶)))
26 olc 399 . . . . 5 (𝐵 < 𝐴 → (𝐴 < 𝐵𝐵 < 𝐴))
2718, 26nsyl 135 . . . 4 (𝜑 → ¬ 𝐵 < 𝐴)
28 biorf 420 . . . 4 𝐵 < 𝐴 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴𝐴 < 𝐶)))
2927, 28syl 17 . . 3 (𝜑 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴𝐴 < 𝐶)))
3025, 29sylibrd 249 . 2 (𝜑 → (𝐵 < 𝐶𝐴 < 𝐶))
3123, 30impbid 202 1 (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  cdif 3556  cun 3557  cop 4159   class class class wbr 4618   × cxp 5077  ccnv 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087
This theorem is referenced by: (None)
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