Theorem pslem 18525

Description: Lemma for psref 18527 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
pslem	⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))

Proof of Theorem pslem

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrel 18522	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
2		brrelex12 5729	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	sylan 581	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		brrelex2 5731	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
5	1, 4	sylan 581	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
6	3, 5	anim12dan 620	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
7		pstr2 18524	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
8		cotr 6112	. . . . . 6 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
9	7, 8	sylib 217	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
10	9	adantr 482	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11		simpr 486	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶))
12		breq12 5154	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
13	12	3adant3 1133	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
14		breq12 5154	. . . . . . . . 9 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
15	14	3adant1 1131	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
16	13, 15	anbi12d 632	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
17		breq12 5154	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
18	17	3adant2 1132	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
19	16, 18	imbi12d 345	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
20	19	spc3gv 3595	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
21	20	3expa 1119	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V) → (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
22	6, 10, 11, 21	syl3c 66	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
23	22	ex 414	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
24		psref2 18523	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
25		asymref2 6119	. . . 4 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
26	25	simplbi 499	. . 3 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) → ∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥)
27		breq12 5154	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
28	27	anidms 568	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
29	28	rspccv 3610	. . 3 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴))
30	24, 26, 29	3syl 18	. 2 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴))
31	3	adantrr 716	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32	25	simprbi 498	. . . . . 6 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
33	24, 32	syl 17	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
34	33	adantr 482	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
35		simpr 486	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
36		breq12 5154	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝐴))
37	36	ancoms 460	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝐴))
38	12, 37	anbi12d 632	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)))
39		eqeq12 2750	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
40	38, 39	imbi12d 345	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))
41	40	spc2gv 3591	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))
42	31, 34, 35, 41	syl3c 66	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → 𝐴 = 𝐵)
43	42	ex 414	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵))
44	23, 30, 43	3jca 1129	1 ⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  ∪ cuni 4909   class class class wbr 5149   I cid 5574  ^◡ccnv 5676   ↾ cres 5679   ∘ ccom 5681  Rel wrel 5682  PosetRelcps 18517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-res 5689  df-ps 18519

This theorem is referenced by:  psdmrn  18526  psref  18527  psasym  18529  pstr  18530