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Theorem swoord1 6465
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 19 . . . 4 (𝜑𝜑)
2 swoord.6 . . . . 5 (𝜑𝐴𝑅𝐵)
3 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
4 difss 3206 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
53, 4eqsstri 3133 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
65ssbri 3979 . . . . 5 (𝐴𝑅𝐵𝐴(𝑋 × 𝑋)𝐵)
7 df-br 3937 . . . . . 6 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
8 opelxp1 4580 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴𝑋)
97, 8sylbi 120 . . . . 5 (𝐴(𝑋 × 𝑋)𝐵𝐴𝑋)
102, 6, 93syl 17 . . . 4 (𝜑𝐴𝑋)
11 swoord.5 . . . 4 (𝜑𝐶𝑋)
12 swoord.4 . . . 4 (𝜑𝐵𝑋)
13 swoer.3 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1413swopolem 4234 . . . 4 ((𝜑 ∧ (𝐴𝑋𝐶𝑋𝐵𝑋)) → (𝐴 < 𝐶 → (𝐴 < 𝐵𝐵 < 𝐶)))
151, 10, 11, 12, 14syl13anc 1219 . . 3 (𝜑 → (𝐴 < 𝐶 → (𝐴 < 𝐵𝐵 < 𝐶)))
163brdifun 6463 . . . . . . 7 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
1710, 12, 16syl2anc 409 . . . . . 6 (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
182, 17mpbid 146 . . . . 5 (𝜑 → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
19 orc 702 . . . . 5 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
2018, 19nsyl 618 . . . 4 (𝜑 → ¬ 𝐴 < 𝐵)
21 biorf 734 . . . 4 𝐴 < 𝐵 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
2220, 21syl 14 . . 3 (𝜑 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
2315, 22sylibrd 168 . 2 (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))
2413swopolem 4234 . . . 4 ((𝜑 ∧ (𝐵𝑋𝐶𝑋𝐴𝑋)) → (𝐵 < 𝐶 → (𝐵 < 𝐴𝐴 < 𝐶)))
251, 12, 11, 10, 24syl13anc 1219 . . 3 (𝜑 → (𝐵 < 𝐶 → (𝐵 < 𝐴𝐴 < 𝐶)))
26 olc 701 . . . . 5 (𝐵 < 𝐴 → (𝐴 < 𝐵𝐵 < 𝐴))
2718, 26nsyl 618 . . . 4 (𝜑 → ¬ 𝐵 < 𝐴)
28 biorf 734 . . . 4 𝐵 < 𝐴 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴𝐴 < 𝐶)))
2927, 28syl 14 . . 3 (𝜑 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴𝐴 < 𝐶)))
3025, 29sylibrd 168 . 2 (𝜑 → (𝐵 < 𝐶𝐴 < 𝐶))
3123, 30impbid 128 1 (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  w3a 963   = wceq 1332  wcel 1481  cdif 3072  cun 3073  cop 3534   class class class wbr 3936   × cxp 4544  ccnv 4545
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-cnv 4554
This theorem is referenced by: (None)
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