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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 3486 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | pweq 4617 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
3 | 2 | eximi 1836 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴) |
4 | bj-snglss 36155 | . . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
5 | sseq2 4009 | . . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴)) | |
6 | 4, 5 | mpbiri 257 | . . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥) |
7 | 6 | eximi 1836 | . . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥) |
8 | vpwex 5376 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V | |
9 | 8 | ssex 5322 | . . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V) |
10 | 9 | exlimiv 1932 | . . . 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 216 | . 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V) |
13 | bj-snglinv 36157 | . . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴} | |
14 | bj-snsetex 36148 | . . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V) | |
15 | 13, 14 | eqeltrid 2836 | . 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V) |
16 | 12, 15 | impbii 208 | 1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ∃wex 1780 ∈ wcel 2105 {cab 2708 Vcvv 3473 ⊆ wss 3949 𝒫 cpw 4603 {csn 4629 sngl bj-csngl 36150 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-pow 5364 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-pw 4605 df-sn 4630 df-pr 4632 df-bj-sngl 36151 |
This theorem is referenced by: bj-tagex 36172 |
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