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Mirrors > Home > MPE Home > Th. List > ersym | Structured version Visualization version GIF version |
Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ersym.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
ersym.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
Ref | Expression |
---|---|
ersym | ⊢ (𝜑 → 𝐵𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersym.2 | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | ersym.1 | . . . . . 6 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
3 | errel 7904 | . . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → Rel 𝑅) |
5 | brrelex12 5295 | . . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
6 | 4, 1, 5 | syl2anc 565 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | brcnvg 5441 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) | |
8 | 7 | ancoms 455 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) |
10 | 1, 9 | mpbird 247 | . 2 ⊢ (𝜑 → 𝐵◡𝑅𝐴) |
11 | df-er 7895 | . . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
12 | 11 | simp3bi 1140 | . . . . 5 ⊢ (𝑅 Er 𝑋 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
13 | 2, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
14 | 13 | unssad 3939 | . . 3 ⊢ (𝜑 → ◡𝑅 ⊆ 𝑅) |
15 | 14 | ssbrd 4827 | . 2 ⊢ (𝜑 → (𝐵◡𝑅𝐴 → 𝐵𝑅𝐴)) |
16 | 10, 15 | mpd 15 | 1 ⊢ (𝜑 → 𝐵𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 Vcvv 3349 ∪ cun 3719 ⊆ wss 3721 class class class wbr 4784 ◡ccnv 5248 dom cdm 5249 ∘ ccom 5253 Rel wrel 5254 Er wer 7892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 df-br 4785 df-opab 4845 df-xp 5255 df-rel 5256 df-cnv 5257 df-er 7895 |
This theorem is referenced by: ercl2 7908 ersymb 7909 ertr2d 7912 ertr3d 7913 ertr4d 7914 erth 7942 erinxp 7972 nqereu 9952 nqerf 9953 1nqenq 9985 qusgrp2 17740 efginvrel2 18346 efgcpbllemb 18374 2idlcpbl 19448 tgptsmscls 22172 |
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