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Mirrors > Home > ILE Home > Th. List > snec | GIF version |
Description: The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
snec.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snec | ⊢ {[𝐴]𝑅} = ({𝐴} / 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snec.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | eceq1 6141 | . . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅) | |
3 | 2 | eqeq2d 2051 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅)) |
4 | 1, 3 | rexsn 3415 | . . 3 ⊢ (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅) |
5 | 4 | abbii 2153 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} |
6 | df-qs 6112 | . 2 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} | |
7 | df-sn 3381 | . 2 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} | |
8 | 5, 6, 7 | 3eqtr4ri 2071 | 1 ⊢ {[𝐴]𝑅} = ({𝐴} / 𝑅) |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 {cab 2026 ∃wrex 2307 Vcvv 2557 {csn 3375 [cec 6104 / cqs 6105 |
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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-ec 6108 df-qs 6112 |
This theorem is referenced by: (None) |
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