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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 3428 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | pweq 4425 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
3 | 2 | eximi 1797 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴) |
4 | bj-snglss 33797 | . . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
5 | sseq2 3884 | . . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴)) | |
6 | 4, 5 | mpbiri 250 | . . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥) |
7 | 6 | eximi 1797 | . . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥) |
8 | vpwex 5131 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V | |
9 | 8 | ssex 5081 | . . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V) |
10 | 9 | exlimiv 1889 | . . . 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 209 | . 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V) |
13 | bj-snglinv 33799 | . . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴} | |
14 | bj-snsetex 33790 | . . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V) | |
15 | 13, 14 | syl5eqel 2871 | . 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V) |
16 | 12, 15 | impbii 201 | 1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ∃wex 1742 ∈ wcel 2050 {cab 2759 Vcvv 3416 ⊆ wss 3830 𝒫 cpw 4422 {csn 4441 sngl bj-csngl 33792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-rex 3095 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-pw 4424 df-sn 4442 df-pr 4444 df-bj-sngl 33793 |
This theorem is referenced by: bj-tagex 33814 |
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