Theorem eusnsn 43618

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 2033	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
2	1	bibi2d 346	. . . 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 4734	. . . . 5 ⊢ (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
5	4	elv 3446	. . . 4 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
6	5	ax-gen 1797	. . 3 ⊢ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
7	3, 6	speivw 1977	. 2 ⊢ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧)
8		eu6 2634	. 2 ⊢ (∃!𝑥{𝑥} = {𝑦} ↔ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧))
9	7, 8	mpbir 234	1 ⊢ ∃!𝑥{𝑥} = {𝑦}

Syntax hints:   ↔ wb 209  ∀wal 1536   = wceq 1538  ∃wex 1781  ∃!weu 2628  Vcvv 3441  {csn 4525

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 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770

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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sn 4526

This theorem is referenced by:  aiotaval  43650