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Mirrors > Home > ILE Home > Th. List > rabsn | GIF version |
Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
Ref | Expression |
---|---|
rabsn | ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2240 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
2 | 1 | pm5.32ri 455 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵)) |
3 | 2 | baib 919 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵)) |
4 | 3 | abbidv 2295 | . 2 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)} = {𝑥 ∣ 𝑥 = 𝐵}) |
5 | df-rab 2464 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)} | |
6 | df-sn 3598 | . 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵} | |
7 | 4, 5, 6 | 3eqtr4g 2235 | 1 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {cab 2163 {crab 2459 {csn 3592 |
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-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-rab 2464 df-sn 3598 |
This theorem is referenced by: unisn3 4445 |
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