Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
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 3506 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | pweq 4555 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
3 | 2 | eximi 1835 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴) |
4 | bj-snglss 34285 | . . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
5 | sseq2 3993 | . . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴)) | |
6 | 4, 5 | mpbiri 260 | . . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥) |
7 | 6 | eximi 1835 | . . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥) |
8 | vpwex 5278 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V | |
9 | 8 | ssex 5225 | . . . . 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 219 | . 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V) |
13 | bj-snglinv 34287 | . . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴} | |
14 | bj-snsetex 34278 | . . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V) | |
15 | 13, 14 | eqeltrid 2917 | . 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V) |
16 | 12, 15 | impbii 211 | 1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 Vcvv 3494 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 sngl bj-csngl 34280 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-pw 4541 df-sn 4568 df-pr 4570 df-bj-sngl 34281 |
This theorem is referenced by: bj-tagex 34302 |
Copyright terms: Public domain | W3C validator |