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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 6115 | . . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → Rel 𝑅) |
5 | brrelex12 4381 | . . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
6 | 4, 1, 5 | syl2anc 391 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | brcnvg 4516 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) | |
8 | 7 | ancoms 255 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) |
9 | 6, 8 | syl 14 | . . 3 ⊢ (𝜑 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) |
10 | 1, 9 | mpbird 156 | . 2 ⊢ (𝜑 → 𝐵◡𝑅𝐴) |
11 | df-er 6106 | . . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
12 | 11 | simp3bi 921 | . . . . 5 ⊢ (𝑅 Er 𝑋 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
13 | 2, 12 | syl 14 | . . . 4 ⊢ (𝜑 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
14 | 13 | unssad 3120 | . . 3 ⊢ (𝜑 → ◡𝑅 ⊆ 𝑅) |
15 | 14 | ssbrd 3805 | . 2 ⊢ (𝜑 → (𝐵◡𝑅𝐴 → 𝐵𝑅𝐴)) |
16 | 10, 15 | mpd 13 | 1 ⊢ (𝜑 → 𝐵𝑅𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∪ cun 2915 ⊆ wss 2917 class class class wbr 3764 ◡ccnv 4344 dom cdm 4345 ∘ ccom 4349 Rel wrel 4350 Er wer 6103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-er 6106 |
This theorem is referenced by: ercl2 6119 ersymb 6120 ertr2d 6123 ertr3d 6124 ertr4d 6125 erth 6150 erinxp 6180 |
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