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Theorem swoer 6164
Description: Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
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)))
Assertion
Ref Expression
swoer (𝜑𝑅 Er 𝑋)
Distinct variable groups:   𝑥,𝑦,𝑧, <   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)

Proof of Theorem swoer
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 swoer.1 . . . . 5 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
2 difss 3097 . . . . 5 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
31, 2eqsstri 3002 . . . 4 𝑅 ⊆ (𝑋 × 𝑋)
4 relxp 4474 . . . 4 Rel (𝑋 × 𝑋)
5 relss 4454 . . . 4 (𝑅 ⊆ (𝑋 × 𝑋) → (Rel (𝑋 × 𝑋) → Rel 𝑅))
63, 4, 5mp2 16 . . 3 Rel 𝑅
76a1i 9 . 2 (𝜑 → Rel 𝑅)
8 simpr 107 . . 3 ((𝜑𝑢𝑅𝑣) → 𝑢𝑅𝑣)
9 orcom 657 . . . . . 6 ((𝑢 < 𝑣𝑣 < 𝑢) ↔ (𝑣 < 𝑢𝑢 < 𝑣))
109a1i 9 . . . . 5 ((𝜑𝑢𝑅𝑣) → ((𝑢 < 𝑣𝑣 < 𝑢) ↔ (𝑣 < 𝑢𝑢 < 𝑣)))
1110notbid 602 . . . 4 ((𝜑𝑢𝑅𝑣) → (¬ (𝑢 < 𝑣𝑣 < 𝑢) ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
123ssbri 3833 . . . . . . 7 (𝑢𝑅𝑣𝑢(𝑋 × 𝑋)𝑣)
1312adantl 266 . . . . . 6 ((𝜑𝑢𝑅𝑣) → 𝑢(𝑋 × 𝑋)𝑣)
14 brxp 4402 . . . . . 6 (𝑢(𝑋 × 𝑋)𝑣 ↔ (𝑢𝑋𝑣𝑋))
1513, 14sylib 131 . . . . 5 ((𝜑𝑢𝑅𝑣) → (𝑢𝑋𝑣𝑋))
161brdifun 6163 . . . . 5 ((𝑢𝑋𝑣𝑋) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
1715, 16syl 14 . . . 4 ((𝜑𝑢𝑅𝑣) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
1815simprd 111 . . . . 5 ((𝜑𝑢𝑅𝑣) → 𝑣𝑋)
1915simpld 109 . . . . 5 ((𝜑𝑢𝑅𝑣) → 𝑢𝑋)
201brdifun 6163 . . . . 5 ((𝑣𝑋𝑢𝑋) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
2118, 19, 20syl2anc 397 . . . 4 ((𝜑𝑢𝑅𝑣) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
2211, 17, 213bitr4d 213 . . 3 ((𝜑𝑢𝑅𝑣) → (𝑢𝑅𝑣𝑣𝑅𝑢))
238, 22mpbid 139 . 2 ((𝜑𝑢𝑅𝑣) → 𝑣𝑅𝑢)
24 simprl 491 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑅𝑣)
2512ad2antrl 467 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢(𝑋 × 𝑋)𝑣)
2614simplbi 263 . . . . . . 7 (𝑢(𝑋 × 𝑋)𝑣𝑢𝑋)
2725, 26syl 14 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑋)
2814simprbi 264 . . . . . . 7 (𝑢(𝑋 × 𝑋)𝑣𝑣𝑋)
2925, 28syl 14 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑣𝑋)
3027, 29, 16syl2anc 397 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
3124, 30mpbid 139 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑣𝑣 < 𝑢))
32 simprr 492 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑣𝑅𝑤)
333brel 4419 . . . . . . . 8 (𝑣𝑅𝑤 → (𝑣𝑋𝑤𝑋))
3433simprd 111 . . . . . . 7 (𝑣𝑅𝑤𝑤𝑋)
3532, 34syl 14 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑤𝑋)
361brdifun 6163 . . . . . 6 ((𝑣𝑋𝑤𝑋) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤𝑤 < 𝑣)))
3729, 35, 36syl2anc 397 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤𝑤 < 𝑣)))
3832, 37mpbid 139 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑣 < 𝑤𝑤 < 𝑣))
39 simpl 106 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝜑)
40 swoer.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
4140swopolem 4069 . . . . . . 7 ((𝜑 ∧ (𝑢𝑋𝑤𝑋𝑣𝑋)) → (𝑢 < 𝑤 → (𝑢 < 𝑣𝑣 < 𝑤)))
4239, 27, 35, 29, 41syl13anc 1148 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢 < 𝑤 → (𝑢 < 𝑣𝑣 < 𝑤)))
4340swopolem 4069 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑋𝑢𝑋𝑣𝑋)) → (𝑤 < 𝑢 → (𝑤 < 𝑣𝑣 < 𝑢)))
4439, 35, 27, 29, 43syl13anc 1148 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑤 < 𝑣𝑣 < 𝑢)))
45 orcom 657 . . . . . . 7 ((𝑣 < 𝑢𝑤 < 𝑣) ↔ (𝑤 < 𝑣𝑣 < 𝑢))
4644, 45syl6ibr 155 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑣 < 𝑢𝑤 < 𝑣)))
4742, 46orim12d 710 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ((𝑢 < 𝑤𝑤 < 𝑢) → ((𝑢 < 𝑣𝑣 < 𝑤) ∨ (𝑣 < 𝑢𝑤 < 𝑣))))
48 or4 698 . . . . 5 (((𝑢 < 𝑣𝑣 < 𝑤) ∨ (𝑣 < 𝑢𝑤 < 𝑣)) ↔ ((𝑢 < 𝑣𝑣 < 𝑢) ∨ (𝑣 < 𝑤𝑤 < 𝑣)))
4947, 48syl6ib 154 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ((𝑢 < 𝑤𝑤 < 𝑢) → ((𝑢 < 𝑣𝑣 < 𝑢) ∨ (𝑣 < 𝑤𝑤 < 𝑣))))
5031, 38, 49mtord 707 . . 3 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑤𝑤 < 𝑢))
511brdifun 6163 . . . 4 ((𝑢𝑋𝑤𝑋) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤𝑤 < 𝑢)))
5227, 35, 51syl2anc 397 . . 3 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤𝑤 < 𝑢)))
5350, 52mpbird 160 . 2 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑅𝑤)
54 swoer.2 . . . . . . 7 ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
5554, 40swopo 4070 . . . . . 6 (𝜑< Po 𝑋)
56 poirr 4071 . . . . . 6 (( < Po 𝑋𝑢𝑋) → ¬ 𝑢 < 𝑢)
5755, 56sylan 271 . . . . 5 ((𝜑𝑢𝑋) → ¬ 𝑢 < 𝑢)
58 pm1.2 683 . . . . 5 ((𝑢 < 𝑢𝑢 < 𝑢) → 𝑢 < 𝑢)
5957, 58nsyl 568 . . . 4 ((𝜑𝑢𝑋) → ¬ (𝑢 < 𝑢𝑢 < 𝑢))
60 simpr 107 . . . . 5 ((𝜑𝑢𝑋) → 𝑢𝑋)
611brdifun 6163 . . . . 5 ((𝑢𝑋𝑢𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢𝑢 < 𝑢)))
6260, 60, 61syl2anc 397 . . . 4 ((𝜑𝑢𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢𝑢 < 𝑢)))
6359, 62mpbird 160 . . 3 ((𝜑𝑢𝑋) → 𝑢𝑅𝑢)
643ssbri 3833 . . . . 5 (𝑢𝑅𝑢𝑢(𝑋 × 𝑋)𝑢)
65 brxp 4402 . . . . . 6 (𝑢(𝑋 × 𝑋)𝑢 ↔ (𝑢𝑋𝑢𝑋))
6665simplbi 263 . . . . 5 (𝑢(𝑋 × 𝑋)𝑢𝑢𝑋)
6764, 66syl 14 . . . 4 (𝑢𝑅𝑢𝑢𝑋)
6867adantl 266 . . 3 ((𝜑𝑢𝑅𝑢) → 𝑢𝑋)
6963, 68impbida 538 . 2 (𝜑 → (𝑢𝑋𝑢𝑅𝑢))
707, 23, 53, 69iserd 6162 1 (𝜑𝑅 Er 𝑋)
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  wss 2944   class class class wbr 3791   Po wpo 4058   × cxp 4370  ccnv 4371  Rel wrel 4377   Er wer 6133
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-po 4060  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-er 6136
This theorem is referenced by: (None)
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