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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 3451 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
2 | 1 | opid 4854 | . . . . . . 7 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}} |
3 | sneq 4600 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
4 | 3 | sneqd 4602 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}}) |
5 | 2, 4 | eqtrid 2785 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}}) |
6 | 5 | sneqd 4602 | . . . . 5 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}}) |
7 | 6 | dmeqd 5865 | . . . 4 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}}) |
8 | 7, 3 | eqeq12d 2749 | . . 3 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴})) |
9 | 1 | dmsnop 6172 | . . 3 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥} |
10 | 8, 9 | vtoclg 3527 | . 2 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) |
11 | 0ex 5268 | . . . . 5 ⊢ ∅ ∈ V | |
12 | 11 | snid 4626 | . . . 4 ⊢ ∅ ∈ {∅} |
13 | dmsn0el 6167 | . . . 4 ⊢ (∅ ∈ {∅} → dom {{∅}} = ∅) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ dom {{∅}} = ∅ |
15 | snprc 4682 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
16 | 15 | biimpi 215 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
17 | 16 | sneqd 4602 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {{𝐴}} = {∅}) |
18 | 17 | sneqd 4602 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {{{𝐴}}} = {{∅}}) |
19 | 18 | dmeqd 5865 | . . 3 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}}) |
20 | 14, 19, 16 | 3eqtr4a 2799 | . 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 1542 ∈ wcel 2107 Vcvv 3447 ∅c0 4286 {csn 4590 ⟨cop 4596 dom cdm 5637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-dm 5647 |
This theorem is referenced by: (None) |
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