![]() |
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 3492 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | pweq 4619 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
3 | 2 | eximi 1832 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴) |
4 | bj-snglss 36953 | . . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴 | |
5 | sseq2 4022 | . . . . . 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 1832 | . . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥) |
8 | vpwex 5383 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V | |
9 | 8 | ssex 5327 | . . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V) |
10 | 9 | exlimiv 1928 | . . . 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 36955 | . . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴} | |
14 | bj-snsetex 36946 | . . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V) | |
15 | 13, 14 | eqeltrid 2843 | . 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 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 sngl bj-csngl 36948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-pw 4607 df-sn 4632 df-pr 4634 df-bj-sngl 36949 |
This theorem is referenced by: bj-tagex 36970 |
Copyright terms: Public domain | W3C validator |