Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eusnsn | Structured version Visualization version GIF version |
Description: There is a unique element of a singleton which is equal to another singleton. (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
eusnsn | ⊢ ∃!𝑥{𝑥} = {𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ2 2033 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
2 | 1 | bibi2d 345 | . . . 4 ⊢ (𝑧 = 𝑦 → (({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))) |
3 | 2 | albidv 1921 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))) |
4 | sneqbg 4774 | . . . . 5 ⊢ (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) | |
5 | 4 | elv 3499 | . . . 4 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦) |
6 | 5 | ax-gen 1796 | . . 3 ⊢ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦) |
7 | 3, 6 | speivw 1977 | . 2 ⊢ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) |
8 | eu6 2659 | . 2 ⊢ (∃!𝑥{𝑥} = {𝑦} ↔ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧)) | |
9 | 7, 8 | mpbir 233 | 1 ⊢ ∃!𝑥{𝑥} = {𝑦} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 = wceq 1537 ∃wex 1780 ∃!weu 2653 Vcvv 3494 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sn 4568 |
This theorem is referenced by: aiotaval 43313 |
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