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Theorem eusnsn 42077
Description: There is a unique element of a singleton which is equal to another singleton. (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
eusnsn ∃!𝑥{𝑥} = {𝑦}
Distinct variable group:   𝑥,𝑦

Proof of Theorem eusnsn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ2 2072 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
21bibi2d 334 . . . 4 (𝑧 = 𝑦 → (({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
32albidv 1963 . . 3 (𝑧 = 𝑦 → (∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
4 sneqbg 4603 . . . . 5 (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
54elv 3401 . . . 4 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
65ax-gen 1839 . . 3 𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
73, 6spei 2358 . 2 𝑧𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧)
8 eu6 2591 . 2 (∃!𝑥{𝑥} = {𝑦} ↔ ∃𝑧𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧))
97, 8mpbir 223 1 ∃!𝑥{𝑥} = {𝑦}
Colors of variables: wff setvar class
Syntax hints:  wb 198  wal 1599   = wceq 1601  wex 1823  ∃!weu 2585  Vcvv 3397  {csn 4397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-v 3399  df-sn 4398
This theorem is referenced by:  aiotaval  42105
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