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Theorem pslem 17409
Description: Lemma for psref 17411 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 17406 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
2 brrelex12 5353 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2sylan 571 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 brrelex2 5355 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
51, 4sylan 571 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
63, 5anim12dan 607 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
7 pstr2 17408 . . . . . 6 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 cotr 5716 . . . . . 6 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
97, 8sylib 209 . . . . 5 (𝑅 ∈ PosetRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109adantr 468 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11 simpr 473 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
12 breq12 4847 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
13123adant3 1155 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
14 breq12 4847 . . . . . . . . 9 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
15143adant1 1153 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1613, 15anbi12d 618 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
17 breq12 4847 . . . . . . . 8 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
18173adant2 1154 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
1916, 18imbi12d 335 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2019spc3gv 3489 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
21203expa 1140 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
226, 10, 11, 21syl3c 66 . . 3 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
2322ex 399 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
24 psref2 17407 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
25 asymref2 5722 . . . 4 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
2625simplbi 487 . . 3 ((𝑅𝑅) = ( I ↾ 𝑅) → ∀𝑥 𝑅𝑥𝑅𝑥)
27 breq12 4847 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
2827anidms 558 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
2928rspccv 3497 . . 3 (∀𝑥 𝑅𝑥𝑅𝑥 → (𝐴 𝑅𝐴𝑅𝐴))
3024, 26, 293syl 18 . 2 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
313adantrr 699 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3225simprbi 486 . . . . . 6 ((𝑅𝑅) = ( I ↾ 𝑅) → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3324, 32syl 17 . . . . 5 (𝑅 ∈ PosetRel → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3433adantr 468 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
35 simpr 473 . . . 4 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → (𝐴𝑅𝐵𝐵𝑅𝐴))
36 breq12 4847 . . . . . . . 8 ((𝑦 = 𝐵𝑥 = 𝐴) → (𝑦𝑅𝑥𝐵𝑅𝐴))
3736ancoms 448 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦𝑅𝑥𝐵𝑅𝐴))
3812, 37anbi12d 618 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
39 eqeq12 2817 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
4038, 39imbi12d 335 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
4140spc2gv 3487 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
4231, 34, 35, 41syl3c 66 . . 3 ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐴)) → 𝐴 = 𝐵)
4342ex 399 . 2 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵))
4423, 30, 433jca 1151 1 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2156  wral 3094  Vcvv 3389  cin 3766  wss 3767   cuni 4628   class class class wbr 4842   I cid 5216  ccnv 5308  cres 5311  ccom 5313  Rel wrel 5314  PosetRelcps 17401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-sep 4973  ax-nul 4981  ax-pr 5094
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ral 3099  df-rex 3100  df-rab 3103  df-v 3391  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4843  df-opab 4905  df-id 5217  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-res 5321  df-ps 17403
This theorem is referenced by:  psdmrn  17410  psref  17411  psasym  17413  pstr  17414
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