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Theorem swoer 7717
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 3715 . . . . 5 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
31, 2eqsstri 3614 . . . 4 𝑅 ⊆ (𝑋 × 𝑋)
4 relxp 5188 . . . 4 Rel (𝑋 × 𝑋)
5 relss 5167 . . . 4 (𝑅 ⊆ (𝑋 × 𝑋) → (Rel (𝑋 × 𝑋) → Rel 𝑅))
63, 4, 5mp2 9 . . 3 Rel 𝑅
76a1i 11 . 2 (𝜑 → Rel 𝑅)
8 simpr 477 . . 3 ((𝜑𝑢𝑅𝑣) → 𝑢𝑅𝑣)
9 orcom 402 . . . . . 6 ((𝑢 < 𝑣𝑣 < 𝑢) ↔ (𝑣 < 𝑢𝑢 < 𝑣))
109a1i 11 . . . . 5 ((𝜑𝑢𝑅𝑣) → ((𝑢 < 𝑣𝑣 < 𝑢) ↔ (𝑣 < 𝑢𝑢 < 𝑣)))
1110notbid 308 . . . 4 ((𝜑𝑢𝑅𝑣) → (¬ (𝑢 < 𝑣𝑣 < 𝑢) ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
123ssbri 4657 . . . . . . 7 (𝑢𝑅𝑣𝑢(𝑋 × 𝑋)𝑣)
1312adantl 482 . . . . . 6 ((𝜑𝑢𝑅𝑣) → 𝑢(𝑋 × 𝑋)𝑣)
14 brxp 5107 . . . . . 6 (𝑢(𝑋 × 𝑋)𝑣 ↔ (𝑢𝑋𝑣𝑋))
1513, 14sylib 208 . . . . 5 ((𝜑𝑢𝑅𝑣) → (𝑢𝑋𝑣𝑋))
161brdifun 7716 . . . . 5 ((𝑢𝑋𝑣𝑋) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
1715, 16syl 17 . . . 4 ((𝜑𝑢𝑅𝑣) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
1815simprd 479 . . . . 5 ((𝜑𝑢𝑅𝑣) → 𝑣𝑋)
1915simpld 475 . . . . 5 ((𝜑𝑢𝑅𝑣) → 𝑢𝑋)
201brdifun 7716 . . . . 5 ((𝑣𝑋𝑢𝑋) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
2118, 19, 20syl2anc 692 . . . 4 ((𝜑𝑢𝑅𝑣) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
2211, 17, 213bitr4d 300 . . 3 ((𝜑𝑢𝑅𝑣) → (𝑢𝑅𝑣𝑣𝑅𝑢))
238, 22mpbid 222 . 2 ((𝜑𝑢𝑅𝑣) → 𝑣𝑅𝑢)
24 simprl 793 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑅𝑣)
2512ad2antrl 763 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢(𝑋 × 𝑋)𝑣)
2614simplbi 476 . . . . . . 7 (𝑢(𝑋 × 𝑋)𝑣𝑢𝑋)
2725, 26syl 17 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑋)
2814simprbi 480 . . . . . . 7 (𝑢(𝑋 × 𝑋)𝑣𝑣𝑋)
2925, 28syl 17 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑣𝑋)
3027, 29, 16syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
3124, 30mpbid 222 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑣𝑣 < 𝑢))
32 simprr 795 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑣𝑅𝑤)
333brel 5128 . . . . . . . 8 (𝑣𝑅𝑤 → (𝑣𝑋𝑤𝑋))
3433simprd 479 . . . . . . 7 (𝑣𝑅𝑤𝑤𝑋)
3532, 34syl 17 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑤𝑋)
361brdifun 7716 . . . . . 6 ((𝑣𝑋𝑤𝑋) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤𝑤 < 𝑣)))
3729, 35, 36syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤𝑤 < 𝑣)))
3832, 37mpbid 222 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑣 < 𝑤𝑤 < 𝑣))
39 simpl 473 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝜑)
40 swoer.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
4140swopolem 5004 . . . . . . 7 ((𝜑 ∧ (𝑢𝑋𝑤𝑋𝑣𝑋)) → (𝑢 < 𝑤 → (𝑢 < 𝑣𝑣 < 𝑤)))
4239, 27, 35, 29, 41syl13anc 1325 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢 < 𝑤 → (𝑢 < 𝑣𝑣 < 𝑤)))
4340swopolem 5004 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑋𝑢𝑋𝑣𝑋)) → (𝑤 < 𝑢 → (𝑤 < 𝑣𝑣 < 𝑢)))
4439, 35, 27, 29, 43syl13anc 1325 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑤 < 𝑣𝑣 < 𝑢)))
45 orcom 402 . . . . . . 7 ((𝑣 < 𝑢𝑤 < 𝑣) ↔ (𝑤 < 𝑣𝑣 < 𝑢))
4644, 45syl6ibr 242 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑣 < 𝑢𝑤 < 𝑣)))
4742, 46orim12d 882 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ((𝑢 < 𝑤𝑤 < 𝑢) → ((𝑢 < 𝑣𝑣 < 𝑤) ∨ (𝑣 < 𝑢𝑤 < 𝑣))))
48 or4 550 . . . . 5 (((𝑢 < 𝑣𝑣 < 𝑤) ∨ (𝑣 < 𝑢𝑤 < 𝑣)) ↔ ((𝑢 < 𝑣𝑣 < 𝑢) ∨ (𝑣 < 𝑤𝑤 < 𝑣)))
4947, 48syl6ib 241 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ((𝑢 < 𝑤𝑤 < 𝑢) → ((𝑢 < 𝑣𝑣 < 𝑢) ∨ (𝑣 < 𝑤𝑤 < 𝑣))))
5031, 38, 49mtord 691 . . 3 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑤𝑤 < 𝑢))
511brdifun 7716 . . . 4 ((𝑢𝑋𝑤𝑋) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤𝑤 < 𝑢)))
5227, 35, 51syl2anc 692 . . 3 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤𝑤 < 𝑢)))
5350, 52mpbird 247 . 2 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑅𝑤)
54 swoer.2 . . . . . . 7 ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
5554, 40swopo 5005 . . . . . 6 (𝜑< Po 𝑋)
56 poirr 5006 . . . . . 6 (( < Po 𝑋𝑢𝑋) → ¬ 𝑢 < 𝑢)
5755, 56sylan 488 . . . . 5 ((𝜑𝑢𝑋) → ¬ 𝑢 < 𝑢)
58 pm1.2 535 . . . . 5 ((𝑢 < 𝑢𝑢 < 𝑢) → 𝑢 < 𝑢)
5957, 58nsyl 135 . . . 4 ((𝜑𝑢𝑋) → ¬ (𝑢 < 𝑢𝑢 < 𝑢))
60 simpr 477 . . . . 5 ((𝜑𝑢𝑋) → 𝑢𝑋)
611brdifun 7716 . . . . 5 ((𝑢𝑋𝑢𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢𝑢 < 𝑢)))
6260, 60, 61syl2anc 692 . . . 4 ((𝜑𝑢𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢𝑢 < 𝑢)))
6359, 62mpbird 247 . . 3 ((𝜑𝑢𝑋) → 𝑢𝑅𝑢)
643ssbri 4657 . . . . 5 (𝑢𝑅𝑢𝑢(𝑋 × 𝑋)𝑢)
65 brxp 5107 . . . . . 6 (𝑢(𝑋 × 𝑋)𝑢 ↔ (𝑢𝑋𝑢𝑋))
6665simplbi 476 . . . . 5 (𝑢(𝑋 × 𝑋)𝑢𝑢𝑋)
6764, 66syl 17 . . . 4 (𝑢𝑅𝑢𝑢𝑋)
6867adantl 482 . . 3 ((𝜑𝑢𝑅𝑢) → 𝑢𝑋)
6963, 68impbida 876 . 2 (𝜑 → (𝑢𝑋𝑢𝑅𝑢))
707, 23, 53, 69iserd 7713 1 (𝜑𝑅 Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  cdif 3552  cun 3553  wss 3555   class class class wbr 4613   Po wpo 4993   × cxp 5072  ccnv 5073  Rel wrel 5079   Er wer 7684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-po 4995  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-er 7687
This theorem is referenced by: (None)
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