Theorem eusnsn 46941

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 2025	. . . . 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 1919	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧) ↔ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
4		sneqbg 4868	. . . . 5 ⊢ (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
5	4	elv 3493	. . . 4 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
6	5	ax-gen 1793	. . 3 ⊢ ∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
7	3, 6	speivw 1973	. 2 ⊢ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧)
8		eu6 2577	. 2 ⊢ (∃!𝑥{𝑥} = {𝑦} ↔ ∃𝑧∀𝑥({𝑥} = {𝑦} ↔ 𝑥 = 𝑧))
9	7, 8	mpbir 231	1 ⊢ ∃!𝑥{𝑥} = {𝑦}

Syntax hints:   ↔ wb 206  ∀wal 1535   = wceq 1537  ∃wex 1777  ∃!weu 2571  Vcvv 3488  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-sn 4649

This theorem is referenced by:  aiotaval  47010