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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-snglex | Structured version Visualization version GIF version | ||
| Description: A class is a set if and only if its singletonization is a set. (Contributed by BJ, 6-Oct-2018.) |
| Ref | Expression |
|---|---|
| bj-snglex | ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isset 3494 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
| 2 | pweq 4614 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
| 3 | 2 | eximi 1835 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴) |
| 4 | bj-snglss 36971 | . . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
| 5 | sseq2 4010 | . . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴)) | |
| 6 | 4, 5 | mpbiri 258 | . . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥) |
| 7 | 6 | eximi 1835 | . . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥) |
| 8 | vpwex 5377 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V | |
| 9 | 8 | ssex 5321 | . . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V) |
| 10 | 9 | exlimiv 1930 | . . . 4 ⊢ (∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V) |
| 11 | 3, 7, 10 | 3syl 18 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → sngl 𝐴 ∈ V) |
| 12 | 1, 11 | sylbi 217 | . 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V) |
| 13 | bj-snglinv 36973 | . . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴} | |
| 14 | bj-snsetex 36964 | . . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V) | |
| 15 | 13, 14 | eqeltrid 2845 | . 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V) |
| 16 | 12, 15 | impbii 209 | 1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 sngl bj-csngl 36966 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-bj-sngl 36967 |
| This theorem is referenced by: bj-tagex 36988 |
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