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Theorem ersym 6714

Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

**Hypotheses**
Ref	Expression
ersym.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ersym.2	⊢ (𝜑 → 𝐴𝑅𝐵)

**Assertion**
Ref	Expression
ersym	⊢ (𝜑 → 𝐵𝑅𝐴)

**Proof of Theorem ersym**
Step	Hyp	Ref	Expression
1		ersym.2	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		ersym.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Er 𝑋)
3		errel 6711	. . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
4	2, 3	syl 14	. . . . 5 ⊢ (𝜑 → Rel 𝑅)
5		brrelex12 4764	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	4, 1, 5	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7		brcnvg 4911	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
8	7	ancoms 268	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
9	6, 8	syl 14	. . 3 ⊢ (𝜑 → (𝐵^◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
10	1, 9	mpbird 167	. 2 ⊢ (𝜑 → 𝐵^◡𝑅𝐴)
11		df-er 6702	. . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
12	11	simp3bi 1040	. . . . 5 ⊢ (𝑅 Er 𝑋 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
13	2, 12	syl 14	. . . 4 ⊢ (𝜑 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
14	13	unssad 3384	. . 3 ⊢ (𝜑 → ^◡𝑅 ⊆ 𝑅)
15	14	ssbrd 4131	. 2 ⊢ (𝜑 → (𝐵^◡𝑅𝐴 → 𝐵𝑅𝐴))
16	10, 15	mpd 13	1 ⊢ (𝜑 → 𝐵𝑅𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1397 ∈ wcel 2202 Vcvv 2802 ∪ cun 3198 ⊆ wss 3200 class class class wbr 4088 ^◡ccnv 4724 dom cdm 4725 ∘ ccom 4729 Rel wrel 4730 Er wer 6699

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-v 2804 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-br 4089 df-opab 4151 df-xp 4731 df-rel 4732 df-cnv 4733 df-er 6702

This theorem is referenced by: ercl2 6715 ersymb 6716 ertr2d 6719 ertr3d 6720 ertr4d 6721 erth 6748 erinxp 6778 qusgrp2 13718 2idlcpblrng 14556

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