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Theorem eusnsn 43557
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 2033 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
21bibi2d 346 . . . 4 (𝑧 = 𝑦 → (({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
32albidv 1921 . . 3 (𝑧 = 𝑦 → (∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
4 sneqbg 4747 . . . . 5 (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
54elv 3474 . . . 4 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
65ax-gen 1797 . . 3 𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
73, 6speivw 1977 . 2 𝑧𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧)
8 eu6 2658 . 2 (∃!𝑥{𝑥} = {𝑦} ↔ ∃𝑧𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧))
97, 8mpbir 234 1 ∃!𝑥{𝑥} = {𝑦}
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536   = wceq 1538  wex 1781  ∃!weu 2652  Vcvv 3469  {csn 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-sn 4540
This theorem is referenced by:  aiotaval  43589
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