Theorem eusnsn 46408

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.

Step	Hyp	Ref	Expression
1		equequ2 2022	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
2	1	bibi2d 342	. . . 4 ⊢ (𝑧 = 𝑦 → (({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
3	2	albidv 1916	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
4		sneqbg 4845	. . . . 5 ⊢ (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
5	4	elv 3477	. . . 4 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
6	5	ax-gen 1790	. . 3 ⊢ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
7	3, 6	speivw 1970	. 2 ⊢ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧)
8		eu6 2564	. 2 ⊢ (∃!𝑥{𝑥} = {𝑦} ↔ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧))
9	7, 8	mpbir 230	1 ⊢ ∃!𝑥{𝑥} = {𝑦}

Syntax hints:   ↔ wb 205  ∀wal 1532   = wceq 1534  ∃wex 1774  ∃!weu 2558  Vcvv 3471  {csn 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-sn 4630

This theorem is referenced by:  aiotaval  46475