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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 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 2 | 1 | opid 4893 | . . . . . . 7 ⊢ 〈𝑥, 𝑥〉 = {{𝑥}} |
| 3 | sneq 4636 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
| 4 | 3 | sneqd 4638 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}}) |
| 5 | 2, 4 | eqtrid 2789 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑥〉 = {{𝐴}}) |
| 6 | 5 | sneqd 4638 | . . . . 5 ⊢ (𝑥 = 𝐴 → {〈𝑥, 𝑥〉} = {{{𝐴}}}) |
| 7 | 6 | dmeqd 5916 | . . . 4 ⊢ (𝑥 = 𝐴 → dom {〈𝑥, 𝑥〉} = dom {{{𝐴}}}) |
| 8 | 7, 3 | eqeq12d 2753 | . . 3 ⊢ (𝑥 = 𝐴 → (dom {〈𝑥, 𝑥〉} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴})) |
| 9 | 1 | dmsnop 6236 | . . 3 ⊢ dom {〈𝑥, 𝑥〉} = {𝑥} |
| 10 | 8, 9 | vtoclg 3554 | . 2 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) |
| 11 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 12 | 11 | snid 4662 | . . . 4 ⊢ ∅ ∈ {∅} |
| 13 | dmsn0el 6231 | . . . 4 ⊢ (∅ ∈ {∅} → dom {{∅}} = ∅) | |
| 14 | 12, 13 | ax-mp 5 | . . 3 ⊢ dom {{∅}} = ∅ |
| 15 | snprc 4717 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 16 | 15 | biimpi 216 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
| 17 | 16 | sneqd 4638 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {{𝐴}} = {∅}) |
| 18 | 17 | sneqd 4638 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {{{𝐴}}} = {{∅}}) |
| 19 | 18 | dmeqd 5916 | . . 3 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}}) |
| 20 | 14, 19, 16 | 3eqtr4a 2803 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 〈cop 4632 dom cdm 5685 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 |
| This theorem is referenced by: (None) |
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