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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 8738 | . . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅) | |
3 | 2 | eqeq2d 2735 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅)) |
4 | 1, 3 | rexsn 4679 | . . 3 ⊢ (∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅) |
5 | 4 | abbii 2794 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} |
6 | df-qs 8706 | . 2 ⊢ ({𝐴} / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑦 = [𝑥]𝑅} | |
7 | df-sn 4622 | . 2 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅} | |
8 | 5, 6, 7 | 3eqtr4ri 2763 | 1 ⊢ {[𝐴]𝑅} = ({𝐴} / 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2701 ∃wrex 3062 Vcvv 3466 {csn 4621 [cec 8698 / cqs 8699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ec 8702 df-qs 8706 |
This theorem is referenced by: (None) |
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