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Theorem pslem 17127
Description: Lemma for psref 17129 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 17124 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
2 brrelex12 5115 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2sylan 488 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 brrelex2 5117 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
51, 4sylan 488 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
63, 5anim12dan 881 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
7 pstr2 17126 . . . . . 6 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 cotr 5467 . . . . . 6 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
97, 8sylib 208 . . . . 5 (𝑅 ∈ PosetRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109adantr 481 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11 simpr 477 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
12 breq12 4618 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
13123adant3 1079 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
14 breq12 4618 . . . . . . . . 9 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
15143adant1 1077 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1613, 15anbi12d 746 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
17 breq12 4618 . . . . . . . 8 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
18173adant2 1078 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
1916, 18imbi12d 334 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2019spc3gv 3284 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
21203expa 1262 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
226, 10, 11, 21syl3c 66 . . 3 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
2322ex 450 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
24 psref2 17125 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
25 asymref2 5472 . . . 4 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
2625simplbi 476 . . 3 ((𝑅𝑅) = ( I ↾ 𝑅) → ∀𝑥 𝑅𝑥𝑅𝑥)
27 breq12 4618 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
2827anidms 676 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
2928rspccv 3292 . . 3 (∀𝑥 𝑅𝑥𝑅𝑥 → (𝐴 𝑅𝐴𝑅𝐴))
3024, 26, 293syl 18 . 2 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
313adantrr 752 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3225simprbi 480 . . . . . 6 ((𝑅𝑅) = ( I ↾ 𝑅) → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3324, 32syl 17 . . . . 5 (𝑅 ∈ PosetRel → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3433adantr 481 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
35 simpr 477 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → (𝐴𝑅𝐵𝐵𝑅𝐴))
36 breq12 4618 . . . . . . . 8 ((𝑦 = 𝐵𝑥 = 𝐴) → (𝑦𝑅𝑥𝐵𝑅𝐴))
3736ancoms 469 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦𝑅𝑥𝐵𝑅𝐴))
3812, 37anbi12d 746 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
39 eqeq12 2634 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
4038, 39imbi12d 334 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
4140spc2gv 3282 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
4231, 34, 35, 41syl3c 66 . . 3 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → 𝐴 = 𝐵)
4342ex 450 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵))
4423, 30, 433jca 1240 1 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cin 3554  wss 3555   cuni 4402   class class class wbr 4613   I cid 4984  ccnv 5073  cres 5076  ccom 5078  Rel wrel 5079  PosetRelcps 17119
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-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-res 5086  df-ps 17121
This theorem is referenced by:  psdmrn  17128  psref  17129  psasym  17131  pstr  17132
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