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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 6569 | . . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → Rel 𝑅) |
5 | brrelex12 4682 | . . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
6 | 4, 1, 5 | syl2anc 411 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | brcnvg 4826 | . . . . 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 6560 | . . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
12 | 11 | simp3bi 1016 | . . . . 5 ⊢ (𝑅 Er 𝑋 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
13 | 2, 12 | syl 14 | . . . 4 ⊢ (𝜑 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
14 | 13 | unssad 3327 | . . 3 ⊢ (𝜑 → ◡𝑅 ⊆ 𝑅) |
15 | 14 | ssbrd 4061 | . 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 1364 ∈ wcel 2160 Vcvv 2752 ∪ cun 3142 ⊆ wss 3144 class class class wbr 4018 ◡ccnv 4643 dom cdm 4644 ∘ ccom 4648 Rel wrel 4649 Er wer 6557 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 df-er 6560 |
This theorem is referenced by: ercl2 6573 ersymb 6574 ertr2d 6577 ertr3d 6578 ertr4d 6579 erth 6606 erinxp 6636 qusgrp2 13070 2idlcpblrng 13855 |
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