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Mirrors > Home > ILE Home > Th. List > dmsnsnsng | GIF version |
Description: The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Ref | Expression |
---|---|
dmsnsnsng | ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2733 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | opid 3781 | . . . . . 6 ⊢ 〈𝑥, 𝑥〉 = {{𝑥}} |
3 | sneq 3592 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
4 | 3 | sneqd 3594 | . . . . . 6 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}}) |
5 | 2, 4 | eqtrid 2215 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑥〉 = {{𝐴}}) |
6 | 5 | sneqd 3594 | . . . 4 ⊢ (𝑥 = 𝐴 → {〈𝑥, 𝑥〉} = {{{𝐴}}}) |
7 | 6 | dmeqd 4811 | . . 3 ⊢ (𝑥 = 𝐴 → dom {〈𝑥, 𝑥〉} = dom {{{𝐴}}}) |
8 | 7, 3 | eqeq12d 2185 | . 2 ⊢ (𝑥 = 𝐴 → (dom {〈𝑥, 𝑥〉} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴})) |
9 | 1 | dmsnop 5082 | . 2 ⊢ dom {〈𝑥, 𝑥〉} = {𝑥} |
10 | 8, 9 | vtoclg 2790 | 1 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 Vcvv 2730 {csn 3581 〈cop 3584 dom cdm 4609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-dm 4619 |
This theorem is referenced by: (None) |
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