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Mirrors > Home > MPE Home > Th. List > snec | Structured version Visualization version 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 8052 | . . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅) | |
3 | 2 | eqeq2d 2835 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅)) |
4 | 1, 3 | rexsn 4445 | . . 3 ⊢ (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅) |
5 | 4 | abbii 2944 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} |
6 | df-qs 8020 | . 2 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} | |
7 | df-sn 4400 | . 2 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} | |
8 | 5, 6, 7 | 3eqtr4ri 2860 | 1 ⊢ {[𝐴]𝑅} = ({𝐴} / 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 {cab 2811 ∃wrex 3118 Vcvv 3414 {csn 4399 [cec 8012 / cqs 8013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-cnv 5354 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-ec 8016 df-qs 8020 |
This theorem is referenced by: (None) |
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