Theorem pslem 18475

Description: Lemma for psref 18477 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 18472	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
2		brrelex12 5668	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	sylan 580	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		brrelex2 5670	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
5	1, 4	sylan 580	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
6	3, 5	anim12dan 619	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
7		pstr2 18474	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
8		cotr 6059	. . . . . 6 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
9	7, 8	sylib 218	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
10	9	adantr 480	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11		simpr 484	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶))
12		breq12 5096	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
13	12	3adant3 1132	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
14		breq12 5096	. . . . . . . . 9 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
15	14	3adant1 1130	. . . . . . . 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 5096	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
18	17	3adant2 1131	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
19	16, 18	imbi12d 344	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
20	19	spc3gv 3559	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
21	20	3expa 1118	. . . 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 412	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
24		psref2 18473	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
25		asymref2 6064	. . . 4 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
26	25	simplbi 497	. . 3 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) → ∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥)
27		breq12 5096	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
28	27	anidms 566	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
29	28	rspccv 3574	. . 3 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴))
30	24, 26, 29	3syl 18	. 2 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴))
31	3	adantrr 717	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32	25	simprbi 496	. . . . . 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 480	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
35		simpr 484	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
36		breq12 5096	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝐴))
37	36	ancoms 458	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝐴))
38	12, 37	anbi12d 632	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)))
39		eqeq12 2748	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
40	38, 39	imbi12d 344	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))
41	40	spc2gv 3555	. . . 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 412	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵))
44	23, 30, 43	3jca 1128	1 ⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4859   class class class wbr 5091   I cid 5510  ^◡ccnv 5615   ↾ cres 5618   ∘ ccom 5620  Rel wrel 5621  PosetRelcps 18467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-res 5628  df-ps 18469

This theorem is referenced by:  psdmrn  18476  psref  18477  psasym  18479  pstr  18480