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Theorem pslem 18630
Description: Lemma for psref 18632 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
pslem (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))

Proof of Theorem pslem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrel 18627 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
2 brrelex12 5741 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2sylan 580 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 brrelex2 5743 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
51, 4sylan 580 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
63, 5anim12dan 619 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
7 pstr2 18629 . . . . . 6 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 cotr 6133 . . . . . 6 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
97, 8sylib 218 . . . . 5 (𝑅 ∈ PosetRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109adantr 480 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11 simpr 484 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
12 breq12 5153 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
13123adant3 1131 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
14 breq12 5153 . . . . . . . . 9 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
15143adant1 1129 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1613, 15anbi12d 632 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
17 breq12 5153 . . . . . . . 8 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
18173adant2 1130 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
1916, 18imbi12d 344 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2019spc3gv 3604 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
21203expa 1117 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
226, 10, 11, 21syl3c 66 . . 3 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
2322ex 412 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
24 psref2 18628 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
25 asymref2 6140 . . . 4 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
2625simplbi 497 . . 3 ((𝑅𝑅) = ( I ↾ 𝑅) → ∀𝑥 𝑅𝑥𝑅𝑥)
27 breq12 5153 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
2827anidms 566 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
2928rspccv 3619 . . 3 (∀𝑥 𝑅𝑥𝑅𝑥 → (𝐴 𝑅𝐴𝑅𝐴))
3024, 26, 293syl 18 . 2 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
313adantrr 717 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3225simprbi 496 . . . . . 6 ((𝑅𝑅) = ( I ↾ 𝑅) → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3324, 32syl 17 . . . . 5 (𝑅 ∈ PosetRel → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3433adantr 480 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
35 simpr 484 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → (𝐴𝑅𝐵𝐵𝑅𝐴))
36 breq12 5153 . . . . . . . 8 ((𝑦 = 𝐵𝑥 = 𝐴) → (𝑦𝑅𝑥𝐵𝑅𝐴))
3736ancoms 458 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦𝑅𝑥𝐵𝑅𝐴))
3812, 37anbi12d 632 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
39 eqeq12 2752 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
4038, 39imbi12d 344 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
4140spc2gv 3600 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
4231, 34, 35, 41syl3c 66 . . 3 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → 𝐴 = 𝐵)
4342ex 412 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵))
4423, 30, 433jca 1127 1 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cin 3962  wss 3963   cuni 4912   class class class wbr 5148   I cid 5582  ccnv 5688  cres 5691  ccom 5693  Rel wrel 5694  PosetRelcps 18622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-res 5701  df-ps 18624
This theorem is referenced by:  psdmrn  18631  psref  18632  psasym  18634  pstr  18635
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