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Theorem pslem 17029
Description: Lemma for psref 17031 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 17026 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
2 brrelex12 5079 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2sylan 487 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 brrelex2 5081 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
51, 4sylan 487 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
63, 5anim12dan 878 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
7 pstr2 17028 . . . . . 6 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 cotr 5427 . . . . . 6 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
97, 8sylib 207 . . . . 5 (𝑅 ∈ PosetRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109adantr 480 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11 simpr 476 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
12 breq12 4588 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
13123adant3 1074 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
14 breq12 4588 . . . . . . . . 9 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
15143adant1 1072 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1613, 15anbi12d 743 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
17 breq12 4588 . . . . . . . 8 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
18173adant2 1073 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
1916, 18imbi12d 333 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2019spc3gv 3271 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
21203expa 1257 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
226, 10, 11, 21syl3c 64 . . 3 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
2322ex 449 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
24 psref2 17027 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
25 asymref2 5432 . . . 4 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
2625simplbi 475 . . 3 ((𝑅𝑅) = ( I ↾ 𝑅) → ∀𝑥 𝑅𝑥𝑅𝑥)
27 breq12 4588 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
2827anidms 675 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
2928rspccv 3279 . . 3 (∀𝑥 𝑅𝑥𝑅𝑥 → (𝐴 𝑅𝐴𝑅𝐴))
3024, 26, 293syl 18 . 2 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
313adantrr 749 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3225simprbi 479 . . . . . 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 476 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → (𝐴𝑅𝐵𝐵𝑅𝐴))
36 breq12 4588 . . . . . . . 8 ((𝑦 = 𝐵𝑥 = 𝐴) → (𝑦𝑅𝑥𝐵𝑅𝐴))
3736ancoms 468 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦𝑅𝑥𝐵𝑅𝐴))
3812, 37anbi12d 743 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
39 eqeq12 2623 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
4038, 39imbi12d 333 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
4140spc2gv 3269 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
4231, 34, 35, 41syl3c 64 . . 3 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → 𝐴 = 𝐵)
4342ex 449 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵))
4423, 30, 433jca 1235 1 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cin 3539  wss 3540   cuni 4372   class class class wbr 4583   I cid 4948  ccnv 5037  cres 5040  ccom 5042  Rel wrel 5043  PosetRelcps 17021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-res 5050  df-ps 17023
This theorem is referenced by:  psdmrn  17030  psref  17031  psasym  17033  pstr  17034
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