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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 6438 | . . . . . 6 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → Rel 𝑅) |
5 | brrelex12 4577 | . . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
6 | 4, 1, 5 | syl2anc 408 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | brcnvg 4720 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) | |
8 | 7 | ancoms 266 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) |
9 | 6, 8 | syl 14 | . . 3 ⊢ (𝜑 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵)) |
10 | 1, 9 | mpbird 166 | . 2 ⊢ (𝜑 → 𝐵◡𝑅𝐴) |
11 | df-er 6429 | . . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
12 | 11 | simp3bi 998 | . . . . 5 ⊢ (𝑅 Er 𝑋 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
13 | 2, 12 | syl 14 | . . . 4 ⊢ (𝜑 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
14 | 13 | unssad 3253 | . . 3 ⊢ (𝜑 → ◡𝑅 ⊆ 𝑅) |
15 | 14 | ssbrd 3971 | . 2 ⊢ (𝜑 → (𝐵◡𝑅𝐴 → 𝐵𝑅𝐴)) |
16 | 10, 15 | mpd 13 | 1 ⊢ (𝜑 → 𝐵𝑅𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 ⊆ wss 3071 class class class wbr 3929 ◡ccnv 4538 dom cdm 4539 ∘ ccom 4543 Rel wrel 4544 Er wer 6426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-er 6429 |
This theorem is referenced by: ercl2 6442 ersymb 6443 ertr2d 6446 ertr3d 6447 ertr4d 6448 erth 6473 erinxp 6503 |
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