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Mirrors > Home > ILE Home > Th. List > ersym | 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 6538 | . . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → Rel 𝑅) |
5 | brrelex12 4661 | . . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
6 | 4, 1, 5 | syl2anc 411 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | brcnvg 4804 | . . . . 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 6529 | . . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
12 | 11 | simp3bi 1014 | . . . . 5 ⊢ (𝑅 Er 𝑋 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
13 | 2, 12 | syl 14 | . . . 4 ⊢ (𝜑 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
14 | 13 | unssad 3312 | . . 3 ⊢ (𝜑 → ◡𝑅 ⊆ 𝑅) |
15 | 14 | ssbrd 4043 | . 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 1353 ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 ⊆ wss 3129 class class class wbr 4000 ◡ccnv 4622 dom cdm 4623 ∘ ccom 4627 Rel wrel 4628 Er wer 6526 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4629 df-rel 4630 df-cnv 4631 df-er 6529 |
This theorem is referenced by: ercl2 6542 ersymb 6543 ertr2d 6546 ertr3d 6547 ertr4d 6548 erth 6573 erinxp 6603 |
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