Theorem erth 8687

Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Hypotheses

Ref	Expression
erth.1	⊢ (𝜑 → 𝑅 Er 𝑋)
erth.2	⊢ (𝜑 → 𝐴 ∈ 𝑋)

Assertion

Ref	Expression
erth	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Proof of Theorem erth

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erth.1	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Er 𝑋)
2	1	ersymb 8647	. . . . . . 7 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
3	2	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
4	1	ertr 8648	. . . . . . 7 ⊢ (𝜑 → ((𝐵𝑅𝐴 ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥))
5	4	impl 455	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
6	3, 5	syldanl 602	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
7	1	ertr 8648	. . . . . 6 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥))
8	7	impl 455	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
9	6, 8	impbida 800	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝑥 ↔ 𝐵𝑅𝑥))
10		vex 3442	. . . . 5 ⊢ 𝑥 ∈ V
11		erth.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ 𝑋)
13		elecg 8677	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
14	10, 12, 13	sylancr 587	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
15		errel 8642	. . . . . . 7 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
16	1, 15	syl 17	. . . . . 6 ⊢ (𝜑 → Rel 𝑅)
17		brrelex2 5676	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
18	16, 17	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
19		elecg 8677	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
20	10, 18, 19	sylancr 587	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
21	9, 14, 20	3bitr4d 311	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
22	21	eqrdv 2732	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
23	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
24	1, 11	erref 8653	. . . . . . 7 ⊢ (𝜑 → 𝐴𝑅𝐴)
25	24	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
26	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ 𝑋)
27		elecg 8677	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
28	26, 26, 27	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
29	25, 28	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
30		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
31	29, 30	eleqtrd 2836	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
32	23, 30	ereldm 8686	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
33	26, 32	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵 ∈ 𝑋)
34		elecg 8677	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
35	26, 33, 34	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
36	31, 35	mpbid 232	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
37	23, 36	ersym 8645	. 2 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
38	22, 37	impbida 800	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438   class class class wbr 5096  Rel wrel 5627   Er wer 8630  [cec 8631

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 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-er 8633  df-ec 8635

This theorem is referenced by:  erth2  8688  erthi  8689  qliftfun  8737  eroveu  8747  eceqoveq  8757  enreceq  10975  prsrlem1  10981  ercpbllem  17467  eqg0el  19110  orbsta  19240  sylow2blem3  19549  qusecsub  19762  frgpnabllem2  19801  rngqipring1  21269  zndvds  21502  qustgpopn  24062  qustgphaus  24065  pi1xfrf  25007  pi1cof  25013  tgjustr  28495  rlocf1  33304  fracfld  33339  qusvscpbl  33381  nsgqusf1olem3  33445  qsnzr  33485  zringfrac  33584  pstmxmet  34003  sconnpi1  35382  topfneec2  36499