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Mirrors > Home > MPE Home > Th. List > rabsn | Structured version Visualization version GIF version |
Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) (Proof shortened by AV, 26-Aug-2022.) |
Ref | Expression |
---|---|
rabsn | ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2902 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
2 | 1 | pm5.32ri 578 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵)) |
3 | 2 | baib 538 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵)) |
4 | 3 | alrimiv 1928 | . 2 ⊢ (𝐵 ∈ 𝐴 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵)) |
5 | rabeqsn 4608 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵)) | |
6 | 4, 5 | sylibr 236 | 1 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {crab 3144 {csn 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-rab 3149 df-sn 4570 |
This theorem is referenced by: unisn3 4861 sylow3lem6 18759 lineunray 33610 pmapat 36901 dia0 38190 nzss 40656 lco0 44489 |
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