Theorem erth 8749

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 8709	. . . . . . 7 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
3	2	biimpa 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
4	1	ertr 8710	. . . . . . 7 ⊢ (𝜑 → ((𝐵𝑅𝐴 ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥))
5	4	impl 460	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
6	3, 5	syldanl 613	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
7	1	ertr 8710	. . . . . 6 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥))
8	7	impl 460	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
9	6, 8	impbida 812	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝑥 ↔ 𝐵𝑅𝑥))
10		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
11		erth.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
12	11	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ 𝑋)
13		elecg 8739	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
14	10, 12, 13	sylancr 598	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
15		errel 8704	. . . . . . 7 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
16	1, 15	syl 18	. . . . . 6 ⊢ (𝜑 → Rel 𝑅)
17		brrelex2 5716	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
18	16, 17	sylan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
19		elecg 8739	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
20	10, 18, 19	sylancr 598	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
21	9, 14, 20	3bitr4d 314	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
22	21	eqrdv 2767	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
23	1	adantr 485	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
24	1, 11	erref 8715	. . . . . . 7 ⊢ (𝜑 → 𝐴𝑅𝐴)
25	24	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
26	11	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ 𝑋)
27		elecg 8739	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
28	26, 26, 27	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
29	25, 28	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
30		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
31	29, 30	eleqtrd 2871	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
32	23, 30	ereldm 8748	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
33	26, 32	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵 ∈ 𝑋)
34		elecg 8739	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
35	26, 33, 34	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
36	31, 35	mpbid 235	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
37	23, 36	ersym 8707	. 2 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
38	22, 37	impbida 812	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   class class class wbr 5113  Rel wrel 5667   Er wer 8691  [cec 8692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-er 8694  df-ec 8696

This theorem is referenced by:  erth2  8750  erthi  8751  qliftfun  8800  eroveu  8810  eceqoveq  8820  enreceq  11051  prsrlem1  11057  ercpbllem  17602  eqg0el  19254  orbsta  19383  sylow2blem3  19692  qusecsub  19905  frgpnabllem2  19944  rngqipring1  21427  qsnzr  21452  zndvds  21668  qustgpopn  24246  qustgphaus  24249  pi1xfrf  25181  pi1cof  25187  tgjustr  28709  erld2  33527  rlocf1  33535  fracfld  33572  qusvscpbl  33614  nsgqusf1olem3  33668  zringfrac  33789  pstmxmet  34232  sconnpi1  35664  topfneec2  36790