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Theorem swoer 6266
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 3115 . . . . 5 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
31, 2eqsstri 3045 . . . 4 𝑅 ⊆ (𝑋 × 𝑋)
4 relxp 4514 . . . 4 Rel (𝑋 × 𝑋)
5 relss 4492 . . . 4 (𝑅 ⊆ (𝑋 × 𝑋) → (Rel (𝑋 × 𝑋) → Rel 𝑅))
63, 4, 5mp2 16 . . 3 Rel 𝑅
76a1i 9 . 2 (𝜑 → Rel 𝑅)
8 simpr 108 . . 3 ((𝜑𝑢𝑅𝑣) → 𝑢𝑅𝑣)
9 orcom 680 . . . . . 6 ((𝑢 < 𝑣𝑣 < 𝑢) ↔ (𝑣 < 𝑢𝑢 < 𝑣))
109a1i 9 . . . . 5 ((𝜑𝑢𝑅𝑣) → ((𝑢 < 𝑣𝑣 < 𝑢) ↔ (𝑣 < 𝑢𝑢 < 𝑣)))
1110notbid 625 . . . 4 ((𝜑𝑢𝑅𝑣) → (¬ (𝑢 < 𝑣𝑣 < 𝑢) ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
123ssbri 3862 . . . . . . 7 (𝑢𝑅𝑣𝑢(𝑋 × 𝑋)𝑣)
1312adantl 271 . . . . . 6 ((𝜑𝑢𝑅𝑣) → 𝑢(𝑋 × 𝑋)𝑣)
14 brxp 4440 . . . . . 6 (𝑢(𝑋 × 𝑋)𝑣 ↔ (𝑢𝑋𝑣𝑋))
1513, 14sylib 120 . . . . 5 ((𝜑𝑢𝑅𝑣) → (𝑢𝑋𝑣𝑋))
161brdifun 6265 . . . . 5 ((𝑢𝑋𝑣𝑋) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
1715, 16syl 14 . . . 4 ((𝜑𝑢𝑅𝑣) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
1815simprd 112 . . . . 5 ((𝜑𝑢𝑅𝑣) → 𝑣𝑋)
1915simpld 110 . . . . 5 ((𝜑𝑢𝑅𝑣) → 𝑢𝑋)
201brdifun 6265 . . . . 5 ((𝑣𝑋𝑢𝑋) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
2118, 19, 20syl2anc 403 . . . 4 ((𝜑𝑢𝑅𝑣) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
2211, 17, 213bitr4d 218 . . 3 ((𝜑𝑢𝑅𝑣) → (𝑢𝑅𝑣𝑣𝑅𝑢))
238, 22mpbid 145 . 2 ((𝜑𝑢𝑅𝑣) → 𝑣𝑅𝑢)
24 simprl 498 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑅𝑣)
2512ad2antrl 474 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢(𝑋 × 𝑋)𝑣)
2614simplbi 268 . . . . . . 7 (𝑢(𝑋 × 𝑋)𝑣𝑢𝑋)
2725, 26syl 14 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑋)
2814simprbi 269 . . . . . . 7 (𝑢(𝑋 × 𝑋)𝑣𝑣𝑋)
2925, 28syl 14 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑣𝑋)
3027, 29, 16syl2anc 403 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
3124, 30mpbid 145 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑣𝑣 < 𝑢))
32 simprr 499 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑣𝑅𝑤)
333brel 4457 . . . . . . . 8 (𝑣𝑅𝑤 → (𝑣𝑋𝑤𝑋))
3433simprd 112 . . . . . . 7 (𝑣𝑅𝑤𝑤𝑋)
3532, 34syl 14 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑤𝑋)
361brdifun 6265 . . . . . 6 ((𝑣𝑋𝑤𝑋) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤𝑤 < 𝑣)))
3729, 35, 36syl2anc 403 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤𝑤 < 𝑣)))
3832, 37mpbid 145 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑣 < 𝑤𝑤 < 𝑣))
39 simpl 107 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝜑)
40 swoer.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
4140swopolem 4105 . . . . . . 7 ((𝜑 ∧ (𝑢𝑋𝑤𝑋𝑣𝑋)) → (𝑢 < 𝑤 → (𝑢 < 𝑣𝑣 < 𝑤)))
4239, 27, 35, 29, 41syl13anc 1174 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢 < 𝑤 → (𝑢 < 𝑣𝑣 < 𝑤)))
4340swopolem 4105 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑋𝑢𝑋𝑣𝑋)) → (𝑤 < 𝑢 → (𝑤 < 𝑣𝑣 < 𝑢)))
4439, 35, 27, 29, 43syl13anc 1174 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑤 < 𝑣𝑣 < 𝑢)))
45 orcom 680 . . . . . . 7 ((𝑣 < 𝑢𝑤 < 𝑣) ↔ (𝑤 < 𝑣𝑣 < 𝑢))
4644, 45syl6ibr 160 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑣 < 𝑢𝑤 < 𝑣)))
4742, 46orim12d 733 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ((𝑢 < 𝑤𝑤 < 𝑢) → ((𝑢 < 𝑣𝑣 < 𝑤) ∨ (𝑣 < 𝑢𝑤 < 𝑣))))
48 or4 721 . . . . 5 (((𝑢 < 𝑣𝑣 < 𝑤) ∨ (𝑣 < 𝑢𝑤 < 𝑣)) ↔ ((𝑢 < 𝑣𝑣 < 𝑢) ∨ (𝑣 < 𝑤𝑤 < 𝑣)))
4947, 48syl6ib 159 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ((𝑢 < 𝑤𝑤 < 𝑢) → ((𝑢 < 𝑣𝑣 < 𝑢) ∨ (𝑣 < 𝑤𝑤 < 𝑣))))
5031, 38, 49mtord 730 . . 3 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑤𝑤 < 𝑢))
511brdifun 6265 . . . 4 ((𝑢𝑋𝑤𝑋) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤𝑤 < 𝑢)))
5227, 35, 51syl2anc 403 . . 3 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤𝑤 < 𝑢)))
5350, 52mpbird 165 . 2 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑅𝑤)
54 swoer.2 . . . . . . 7 ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
5554, 40swopo 4106 . . . . . 6 (𝜑< Po 𝑋)
56 poirr 4107 . . . . . 6 (( < Po 𝑋𝑢𝑋) → ¬ 𝑢 < 𝑢)
5755, 56sylan 277 . . . . 5 ((𝜑𝑢𝑋) → ¬ 𝑢 < 𝑢)
58 pm1.2 706 . . . . 5 ((𝑢 < 𝑢𝑢 < 𝑢) → 𝑢 < 𝑢)
5957, 58nsyl 591 . . . 4 ((𝜑𝑢𝑋) → ¬ (𝑢 < 𝑢𝑢 < 𝑢))
60 simpr 108 . . . . 5 ((𝜑𝑢𝑋) → 𝑢𝑋)
611brdifun 6265 . . . . 5 ((𝑢𝑋𝑢𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢𝑢 < 𝑢)))
6260, 60, 61syl2anc 403 . . . 4 ((𝜑𝑢𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢𝑢 < 𝑢)))
6359, 62mpbird 165 . . 3 ((𝜑𝑢𝑋) → 𝑢𝑅𝑢)
643ssbri 3862 . . . . 5 (𝑢𝑅𝑢𝑢(𝑋 × 𝑋)𝑢)
65 brxp 4440 . . . . . 6 (𝑢(𝑋 × 𝑋)𝑢 ↔ (𝑢𝑋𝑢𝑋))
6665simplbi 268 . . . . 5 (𝑢(𝑋 × 𝑋)𝑢𝑢𝑋)
6764, 66syl 14 . . . 4 (𝑢𝑅𝑢𝑢𝑋)
6867adantl 271 . . 3 ((𝜑𝑢𝑅𝑢) → 𝑢𝑋)
6963, 68impbida 561 . 2 (𝜑 → (𝑢𝑋𝑢𝑅𝑢))
707, 23, 53, 69iserd 6264 1 (𝜑𝑅 Er 𝑋)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3a 922   = wceq 1287  wcel 1436  cdif 2985  cun 2986  wss 2988   class class class wbr 3820   Po wpo 4094   × cxp 4408  ccnv 4409  Rel wrel 4415   Er wer 6235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-pow 3983  ax-pr 4009
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-po 4096  df-xp 4416  df-rel 4417  df-cnv 4418  df-co 4419  df-dm 4420  df-er 6238
This theorem is referenced by: (None)
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