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Theorem swoer 7977
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 3899 . . . . 5 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
31, 2eqsstri 3795 . . . 4 𝑅 ⊆ (𝑋 × 𝑋)
4 relxp 5295 . . . 4 Rel (𝑋 × 𝑋)
5 relss 5376 . . . 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 896 . . . . . 6 ((𝑢 < 𝑣𝑣 < 𝑢) ↔ (𝑣 < 𝑢𝑢 < 𝑣))
109a1i 11 . . . . 5 ((𝜑𝑢𝑅𝑣) → ((𝑢 < 𝑣𝑣 < 𝑢) ↔ (𝑣 < 𝑢𝑢 < 𝑣)))
1110notbid 309 . . . 4 ((𝜑𝑢𝑅𝑣) → (¬ (𝑢 < 𝑣𝑣 < 𝑢) ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
123ssbri 4854 . . . . . . 7 (𝑢𝑅𝑣𝑢(𝑋 × 𝑋)𝑣)
1312adantl 473 . . . . . 6 ((𝜑𝑢𝑅𝑣) → 𝑢(𝑋 × 𝑋)𝑣)
14 brxp 5323 . . . . . 6 (𝑢(𝑋 × 𝑋)𝑣 ↔ (𝑢𝑋𝑣𝑋))
1513, 14sylib 209 . . . . 5 ((𝜑𝑢𝑅𝑣) → (𝑢𝑋𝑣𝑋))
161brdifun 7976 . . . . 5 ((𝑢𝑋𝑣𝑋) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
1715, 16syl 17 . . . 4 ((𝜑𝑢𝑅𝑣) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
1815simprd 489 . . . . 5 ((𝜑𝑢𝑅𝑣) → 𝑣𝑋)
1915simpld 488 . . . . 5 ((𝜑𝑢𝑅𝑣) → 𝑢𝑋)
201brdifun 7976 . . . . 5 ((𝑣𝑋𝑢𝑋) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
2118, 19, 20syl2anc 579 . . . 4 ((𝜑𝑢𝑅𝑣) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢𝑢 < 𝑣)))
2211, 17, 213bitr4d 302 . . 3 ((𝜑𝑢𝑅𝑣) → (𝑢𝑅𝑣𝑣𝑅𝑢))
238, 22mpbid 223 . 2 ((𝜑𝑢𝑅𝑣) → 𝑣𝑅𝑢)
24 simprl 787 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑅𝑣)
2512ad2antrl 719 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢(𝑋 × 𝑋)𝑣)
2614simplbi 491 . . . . . . 7 (𝑢(𝑋 × 𝑋)𝑣𝑢𝑋)
2725, 26syl 17 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑋)
2814simprbi 490 . . . . . . 7 (𝑢(𝑋 × 𝑋)𝑣𝑣𝑋)
2925, 28syl 17 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑣𝑋)
3027, 29, 16syl2anc 579 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣𝑣 < 𝑢)))
3124, 30mpbid 223 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑣𝑣 < 𝑢))
32 simprr 789 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑣𝑅𝑤)
333brel 5336 . . . . . . . 8 (𝑣𝑅𝑤 → (𝑣𝑋𝑤𝑋))
3433simprd 489 . . . . . . 7 (𝑣𝑅𝑤𝑤𝑋)
3532, 34syl 17 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑤𝑋)
361brdifun 7976 . . . . . 6 ((𝑣𝑋𝑤𝑋) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤𝑤 < 𝑣)))
3729, 35, 36syl2anc 579 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤𝑤 < 𝑣)))
3832, 37mpbid 223 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑣 < 𝑤𝑤 < 𝑣))
39 simpl 474 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝜑)
40 swoer.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
4140swopolem 5207 . . . . . . 7 ((𝜑 ∧ (𝑢𝑋𝑤𝑋𝑣𝑋)) → (𝑢 < 𝑤 → (𝑢 < 𝑣𝑣 < 𝑤)))
4239, 27, 35, 29, 41syl13anc 1491 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢 < 𝑤 → (𝑢 < 𝑣𝑣 < 𝑤)))
4340swopolem 5207 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑋𝑢𝑋𝑣𝑋)) → (𝑤 < 𝑢 → (𝑤 < 𝑣𝑣 < 𝑢)))
4439, 35, 27, 29, 43syl13anc 1491 . . . . . . 7 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑤 < 𝑣𝑣 < 𝑢)))
45 orcom 896 . . . . . . 7 ((𝑣 < 𝑢𝑤 < 𝑣) ↔ (𝑤 < 𝑣𝑣 < 𝑢))
4644, 45syl6ibr 243 . . . . . 6 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑣 < 𝑢𝑤 < 𝑣)))
4742, 46orim12d 987 . . . . 5 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ((𝑢 < 𝑤𝑤 < 𝑢) → ((𝑢 < 𝑣𝑣 < 𝑤) ∨ (𝑣 < 𝑢𝑤 < 𝑣))))
48 or4 950 . . . . 5 (((𝑢 < 𝑣𝑣 < 𝑤) ∨ (𝑣 < 𝑢𝑤 < 𝑣)) ↔ ((𝑢 < 𝑣𝑣 < 𝑢) ∨ (𝑣 < 𝑤𝑤 < 𝑣)))
4947, 48syl6ib 242 . . . 4 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ((𝑢 < 𝑤𝑤 < 𝑢) → ((𝑢 < 𝑣𝑣 < 𝑢) ∨ (𝑣 < 𝑤𝑤 < 𝑣))))
5031, 38, 49mtord 903 . . 3 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑤𝑤 < 𝑢))
511brdifun 7976 . . . 4 ((𝑢𝑋𝑤𝑋) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤𝑤 < 𝑢)))
5227, 35, 51syl2anc 579 . . 3 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤𝑤 < 𝑢)))
5350, 52mpbird 248 . 2 ((𝜑 ∧ (𝑢𝑅𝑣𝑣𝑅𝑤)) → 𝑢𝑅𝑤)
54 swoer.2 . . . . . . 7 ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
5554, 40swopo 5208 . . . . . 6 (𝜑< Po 𝑋)
56 poirr 5209 . . . . . 6 (( < Po 𝑋𝑢𝑋) → ¬ 𝑢 < 𝑢)
5755, 56sylan 575 . . . . 5 ((𝜑𝑢𝑋) → ¬ 𝑢 < 𝑢)
58 pm1.2 927 . . . . 5 ((𝑢 < 𝑢𝑢 < 𝑢) → 𝑢 < 𝑢)
5957, 58nsyl 137 . . . 4 ((𝜑𝑢𝑋) → ¬ (𝑢 < 𝑢𝑢 < 𝑢))
60 simpr 477 . . . . 5 ((𝜑𝑢𝑋) → 𝑢𝑋)
611brdifun 7976 . . . . 5 ((𝑢𝑋𝑢𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢𝑢 < 𝑢)))
6260, 60, 61syl2anc 579 . . . 4 ((𝜑𝑢𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢𝑢 < 𝑢)))
6359, 62mpbird 248 . . 3 ((𝜑𝑢𝑋) → 𝑢𝑅𝑢)
643ssbri 4854 . . . . 5 (𝑢𝑅𝑢𝑢(𝑋 × 𝑋)𝑢)
65 brxp 5323 . . . . . 6 (𝑢(𝑋 × 𝑋)𝑢 ↔ (𝑢𝑋𝑢𝑋))
6665simplbi 491 . . . . 5 (𝑢(𝑋 × 𝑋)𝑢𝑢𝑋)
6764, 66syl 17 . . . 4 (𝑢𝑅𝑢𝑢𝑋)
6867adantl 473 . . 3 ((𝜑𝑢𝑅𝑢) → 𝑢𝑋)
6963, 68impbida 835 . 2 (𝜑 → (𝑢𝑋𝑢𝑅𝑢))
707, 23, 53, 69iserd 7973 1 (𝜑𝑅 Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  cdif 3729  cun 3730  wss 3732   class class class wbr 4809   Po wpo 5196   × cxp 5275  ccnv 5276  Rel wrel 5282   Er wer 7944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-po 5198  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-er 7947
This theorem is referenced by: (None)
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