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Theorem swoord2 6166
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 19 . . . 4 (𝜑𝜑)
2 swoord.5 . . . 4 (𝜑𝐶𝑋)
3 swoord.6 . . . . 5 (𝜑𝐴𝑅𝐵)
4 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
5 difss 3097 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
64, 5eqsstri 3002 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
76ssbri 3833 . . . . 5 (𝐴𝑅𝐵𝐴(𝑋 × 𝑋)𝐵)
8 df-br 3792 . . . . . 6 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
9 opelxp1 4404 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴𝑋)
108, 9sylbi 118 . . . . 5 (𝐴(𝑋 × 𝑋)𝐵𝐴𝑋)
113, 7, 103syl 17 . . . 4 (𝜑𝐴𝑋)
12 swoord.4 . . . 4 (𝜑𝐵𝑋)
13 swoer.3 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1413swopolem 4069 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
151, 2, 11, 12, 14syl13anc 1148 . . 3 (𝜑 → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
16 idd 21 . . . 4 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐵))
174brdifun 6163 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
1811, 12, 17syl2anc 397 . . . . . . 7 (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
193, 18mpbid 139 . . . . . 6 (𝜑 → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
20 olc 642 . . . . . 6 (𝐵 < 𝐴 → (𝐴 < 𝐵𝐵 < 𝐴))
2119, 20nsyl 568 . . . . 5 (𝜑 → ¬ 𝐵 < 𝐴)
2221pm2.21d 559 . . . 4 (𝜑 → (𝐵 < 𝐴𝐶 < 𝐵))
2316, 22jaod 647 . . 3 (𝜑 → ((𝐶 < 𝐵𝐵 < 𝐴) → 𝐶 < 𝐵))
2415, 23syld 44 . 2 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
2513swopolem 4069 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐵𝑋𝐴𝑋)) → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
261, 2, 12, 11, 25syl13anc 1148 . . 3 (𝜑 → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
27 idd 21 . . . 4 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐴))
28 orc 643 . . . . . 6 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
2919, 28nsyl 568 . . . . 5 (𝜑 → ¬ 𝐴 < 𝐵)
3029pm2.21d 559 . . . 4 (𝜑 → (𝐴 < 𝐵𝐶 < 𝐴))
3127, 30jaod 647 . . 3 (𝜑 → ((𝐶 < 𝐴𝐴 < 𝐵) → 𝐶 < 𝐴))
3226, 31syld 44 . 2 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐴))
3324, 32impbid 124 1 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  w3a 896   = wceq 1259  wcel 1409  cdif 2941  cun 2942  cop 3405   class class class wbr 3791   × cxp 4370  ccnv 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380
This theorem is referenced by: (None)
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