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Mirrors > Home > MPE Home > Th. List > dmsnsnsn | Structured version Visualization version GIF version |
Description: The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnsnsn | ⊢ dom {{{𝐴}}} = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3476 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
2 | 1 | opid 4892 | . . . . . . 7 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}} |
3 | sneq 4637 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
4 | 3 | sneqd 4639 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}}) |
5 | 2, 4 | eqtrid 2782 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}}) |
6 | 5 | sneqd 4639 | . . . . 5 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}}) |
7 | 6 | dmeqd 5904 | . . . 4 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}}) |
8 | 7, 3 | eqeq12d 2746 | . . 3 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴})) |
9 | 1 | dmsnop 6214 | . . 3 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥} |
10 | 8, 9 | vtoclg 3541 | . 2 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) |
11 | 0ex 5306 | . . . . 5 ⊢ ∅ ∈ V | |
12 | 11 | snid 4663 | . . . 4 ⊢ ∅ ∈ {∅} |
13 | dmsn0el 6209 | . . . 4 ⊢ (∅ ∈ {∅} → dom {{∅}} = ∅) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ dom {{∅}} = ∅ |
15 | snprc 4720 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
16 | 15 | biimpi 215 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
17 | 16 | sneqd 4639 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {{𝐴}} = {∅}) |
18 | 17 | sneqd 4639 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {{{𝐴}}} = {{∅}}) |
19 | 18 | dmeqd 5904 | . . 3 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}}) |
20 | 14, 19, 16 | 3eqtr4a 2796 | . 2 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) |
21 | 10, 20 | pm2.61i 182 | 1 ⊢ dom {{{𝐴}}} = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∅c0 4321 {csn 4627 ⟨cop 4633 dom cdm 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-dm 5685 |
This theorem is referenced by: (None) |
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