Theorem pslem 17206

Description: Lemma for psref 17208 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 17203	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
2		brrelex12 5155	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	sylan 488	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		brrelex2 5157	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
5	1, 4	sylan 488	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
6	3, 5	anim12dan 882	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
7		pstr2 17205	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
8		cotr 5508	. . . . . 6 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
9	7, 8	sylib 208	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
10	9	adantr 481	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11		simpr 477	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶))
12		breq12 4658	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
13	12	3adant3 1081	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
14		breq12 4658	. . . . . . . . 9 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
15	14	3adant1 1079	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
16	13, 15	anbi12d 747	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
17		breq12 4658	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
18	17	3adant2 1080	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
19	16, 18	imbi12d 334	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
20	19	spc3gv 3298	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
21	20	3expa 1265	. . . 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 450	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
24		psref2 17204	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
25		asymref2 5513	. . . 4 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
26	25	simplbi 476	. . 3 ⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) → ∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥)
27		breq12 4658	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
28	27	anidms 677	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
29	28	rspccv 3306	. . 3 ⊢ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴))
30	24, 26, 29	3syl 18	. 2 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴))
31	3	adantrr 753	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32	25	simprbi 480	. . . . . 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 481	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
35		simpr 477	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
36		breq12 4658	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝐴))
37	36	ancoms 469	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦𝑅𝑥 ↔ 𝐵𝑅𝐴))
38	12, 37	anbi12d 747	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)))
39		eqeq12 2635	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
40	38, 39	imbi12d 334	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))
41	40	spc2gv 3296	. . . 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 450	. 2 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵))
44	23, 30, 43	3jca 1242	1 ⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∪ cuni 4436   class class class wbr 4653   I cid 5023  ^◡ccnv 5113   ↾ cres 5116   ∘ ccom 5118  Rel wrel 5119  PosetRelcps 17198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-ps 17200

This theorem is referenced by:  psdmrn  17207  psref  17208  psasym  17210  pstr  17211