Theorem erth 6665

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		simpl 109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜑)
2		erth.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 Er 𝑋)
3	2	ersymb 6633	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
4	3	biimpa 296	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
5	1, 4	jca 306	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝜑 ∧ 𝐵𝑅𝐴))
6	2	ertr 6634	. . . . . . 7 ⊢ (𝜑 → ((𝐵𝑅𝐴 ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥))
7	6	impl 380	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
8	5, 7	sylan 283	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
9	2	ertr 6634	. . . . . 6 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥))
10	9	impl 380	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
11	8, 10	impbida 596	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝑥 ↔ 𝐵𝑅𝑥))
12		vex 2774	. . . . 5 ⊢ 𝑥 ∈ V
13		erth.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
14	13	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ 𝑋)
15		elecg 6659	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
16	12, 14, 15	sylancr 414	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
17		errel 6628	. . . . . . 7 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
18	2, 17	syl 14	. . . . . 6 ⊢ (𝜑 → Rel 𝑅)
19		brrelex2 4715	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
20	18, 19	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
21		elecg 6659	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
22	12, 20, 21	sylancr 414	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
23	11, 16, 22	3bitr4d 220	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
24	23	eqrdv 2202	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
25	2	adantr 276	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
26	2, 13	erref 6639	. . . . . . 7 ⊢ (𝜑 → 𝐴𝑅𝐴)
27	26	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
28	13	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ 𝑋)
29		elecg 6659	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
30	28, 28, 29	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
31	27, 30	mpbird 167	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
32		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
33	31, 32	eleqtrd 2283	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
34	25, 32	ereldm 6664	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
35	28, 34	mpbid 147	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵 ∈ 𝑋)
36		elecg 6659	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
37	28, 35, 36	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
38	33, 37	mpbid 147	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
39	25, 38	ersym 6631	. 2 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
40	24, 39	impbida 596	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  Vcvv 2771   class class class wbr 4043  Rel wrel 4679   Er wer 6616  [cec 6617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-er 6619  df-ec 6621

This theorem is referenced by:  erth2  6666  erthi  6667  qliftfun  6703  eroveu  6712  th3qlem1  6723  enqeceq  7471  enq0eceq  7549  nnnq0lem1  7558  enreceq  7848  prsrlem1  7854  ercpbllemg  13133  eqg0el  13536  qusecsub  13638  zndvds  14382