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Theorem swoord2 7928
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 3888 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
64, 5eqsstri 3784 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
76ssbri 4831 . . . . 5 (𝐴𝑅𝐵𝐴(𝑋 × 𝑋)𝐵)
8 df-br 4787 . . . . . 6 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
9 opelxp1 5290 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴𝑋)
108, 9sylbi 207 . . . . 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 5179 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
151, 2, 11, 12, 14syl13anc 1478 . . 3 (𝜑 → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
16 idd 24 . . . 4 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐵))
174brdifun 7925 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
1811, 12, 17syl2anc 573 . . . . . . 7 (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
193, 18mpbid 222 . . . . . 6 (𝜑 → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
20 olc 855 . . . . . 6 (𝐵 < 𝐴 → (𝐴 < 𝐵𝐵 < 𝐴))
2119, 20nsyl 137 . . . . 5 (𝜑 → ¬ 𝐵 < 𝐴)
2221pm2.21d 119 . . . 4 (𝜑 → (𝐵 < 𝐴𝐶 < 𝐵))
2316, 22jaod 846 . . 3 (𝜑 → ((𝐶 < 𝐵𝐵 < 𝐴) → 𝐶 < 𝐵))
2415, 23syld 47 . 2 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
2513swopolem 5179 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐵𝑋𝐴𝑋)) → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
261, 2, 12, 11, 25syl13anc 1478 . . 3 (𝜑 → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
27 idd 24 . . . 4 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐴))
28 orc 854 . . . . . 6 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
2919, 28nsyl 137 . . . . 5 (𝜑 → ¬ 𝐴 < 𝐵)
3029pm2.21d 119 . . . 4 (𝜑 → (𝐴 < 𝐵𝐶 < 𝐴))
3127, 30jaod 846 . . 3 (𝜑 → ((𝐶 < 𝐴𝐴 < 𝐵) → 𝐶 < 𝐴))
3226, 31syld 47 . 2 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐴))
3324, 32impbid 202 1 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 834  w3a 1071   = wceq 1631  wcel 2145  cdif 3720  cun 3721  cop 4322   class class class wbr 4786   × cxp 5247  ccnv 5248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257
This theorem is referenced by: (None)
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