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Theorem swoord2 6512
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 3234 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
64, 5eqsstri 3160 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
76ssbri 4010 . . . . 5 (𝐴𝑅𝐵𝐴(𝑋 × 𝑋)𝐵)
8 df-br 3968 . . . . . 6 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
9 opelxp1 4622 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴𝑋)
108, 9sylbi 120 . . . . 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 4267 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
151, 2, 11, 12, 14syl13anc 1222 . . 3 (𝜑 → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
16 idd 21 . . . 4 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐵))
174brdifun 6509 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
1811, 12, 17syl2anc 409 . . . . . . 7 (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
193, 18mpbid 146 . . . . . 6 (𝜑 → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
20 olc 701 . . . . . 6 (𝐵 < 𝐴 → (𝐴 < 𝐵𝐵 < 𝐴))
2119, 20nsyl 618 . . . . 5 (𝜑 → ¬ 𝐵 < 𝐴)
2221pm2.21d 609 . . . 4 (𝜑 → (𝐵 < 𝐴𝐶 < 𝐵))
2316, 22jaod 707 . . 3 (𝜑 → ((𝐶 < 𝐵𝐵 < 𝐴) → 𝐶 < 𝐵))
2415, 23syld 45 . 2 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
2513swopolem 4267 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐵𝑋𝐴𝑋)) → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
261, 2, 12, 11, 25syl13anc 1222 . . 3 (𝜑 → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
27 idd 21 . . . 4 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐴))
28 orc 702 . . . . . 6 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
2919, 28nsyl 618 . . . . 5 (𝜑 → ¬ 𝐴 < 𝐵)
3029pm2.21d 609 . . . 4 (𝜑 → (𝐴 < 𝐵𝐶 < 𝐴))
3127, 30jaod 707 . . 3 (𝜑 → ((𝐶 < 𝐴𝐴 < 𝐵) → 𝐶 < 𝐴))
3226, 31syld 45 . 2 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐴))
3324, 32impbid 128 1 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  w3a 963   = wceq 1335  wcel 2128  cdif 3099  cun 3100  cop 3564   class class class wbr 3967   × cxp 4586  ccnv 4587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-cnv 4596
This theorem is referenced by: (None)
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