Theorem erth 6545

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 108	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜑)
2		erth.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 Er 𝑋)
3	2	ersymb 6515	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
4	3	biimpa 294	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
5	1, 4	jca 304	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝜑 ∧ 𝐵𝑅𝐴))
6	2	ertr 6516	. . . . . . 7 ⊢ (𝜑 → ((𝐵𝑅𝐴 ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥))
7	6	impl 378	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
8	5, 7	sylan 281	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
9	2	ertr 6516	. . . . . 6 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥))
10	9	impl 378	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
11	8, 10	impbida 586	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝑥 ↔ 𝐵𝑅𝑥))
12		vex 2729	. . . . 5 ⊢ 𝑥 ∈ V
13		erth.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
14	13	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ 𝑋)
15		elecg 6539	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
16	12, 14, 15	sylancr 411	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
17		errel 6510	. . . . . . 7 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
18	2, 17	syl 14	. . . . . 6 ⊢ (𝜑 → Rel 𝑅)
19		brrelex2 4645	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
20	18, 19	sylan 281	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
21		elecg 6539	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
22	12, 20, 21	sylancr 411	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
23	11, 16, 22	3bitr4d 219	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
24	23	eqrdv 2163	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
25	2	adantr 274	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
26	2, 13	erref 6521	. . . . . . 7 ⊢ (𝜑 → 𝐴𝑅𝐴)
27	26	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
28	13	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ 𝑋)
29		elecg 6539	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
30	28, 28, 29	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
31	27, 30	mpbird 166	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
32		simpr 109	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
33	31, 32	eleqtrd 2245	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
34	25, 32	ereldm 6544	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
35	28, 34	mpbid 146	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵 ∈ 𝑋)
36		elecg 6539	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
37	28, 35, 36	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
38	33, 37	mpbid 146	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
39	25, 38	ersym 6513	. 2 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
40	24, 39	impbida 586	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  Vcvv 2726   class class class wbr 3982  Rel wrel 4609   Er wer 6498  [cec 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-er 6501  df-ec 6503

This theorem is referenced by:  erth2  6546  erthi  6547  qliftfun  6583  eroveu  6592  th3qlem1  6603  enqeceq  7300  enq0eceq  7378  nnnq0lem1  7387  enreceq  7677  prsrlem1  7683