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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 3453 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | pweq 4513 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
3 | 2 | eximi 1836 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴) |
4 | bj-snglss 34406 | . . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
5 | sseq2 3941 | . . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴)) | |
6 | 4, 5 | mpbiri 261 | . . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥) |
7 | 6 | eximi 1836 | . . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥) |
8 | vpwex 5243 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V | |
9 | 8 | ssex 5189 | . . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V) |
10 | 9 | exlimiv 1931 | . . . 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 220 | . 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V) |
13 | bj-snglinv 34408 | . . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴} | |
14 | bj-snsetex 34399 | . . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V) | |
15 | 13, 14 | eqeltrid 2894 | . 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V) |
16 | 12, 15 | impbii 212 | 1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 {csn 4525 sngl bj-csngl 34401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-bj-sngl 34402 |
This theorem is referenced by: bj-tagex 34423 |
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