Theorem eusnsn 47708

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 2054	. . . . 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 1948	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
4		sneqbg 4807	. . . . 5 ⊢ (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
5	4	elv 3458	. . . 4 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
6	5	ax-gen 1823	. . 3 ⊢ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
7	3, 6	speivw 2001	. 2 ⊢ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧)
8		eu6 2600	. 2 ⊢ (∃!𝑥{𝑥} = {𝑦} ↔ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧))
9	7, 8	mpbir 234	1 ⊢ ∃!𝑥{𝑥} = {𝑦}

Syntax hints:   ↔ wb 209  ∀wal 1566   = wceq 1568  ∃wex 1807  ∃!weu 2594  Vcvv 3453  {csn 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-sn 4589

This theorem is referenced by:  aiotaval  47777