Theorem mosneq 4796

Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3663. (Contributed by BJ, 24-Sep-2022.)

Assertion

Ref	Expression
mosneq	⊢ ∃*𝑥{𝑥} = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem mosneq

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2756	. . . 4 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦})
2		vex 3442	. . . . 5 ⊢ 𝑥 ∈ V
3	2	sneqr 4794	. . . 4 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
4	1, 3	syl 17	. . 3 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
5	4	gen2 1797	. 2 ⊢ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
6		sneq 4588	. . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
7	6	eqeq1d 2736	. . 3 ⊢ (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴))
8	7	mo4 2564	. 2 ⊢ (∃*𝑥{𝑥} = 𝐴 ↔ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦))
9	5, 8	mpbir 231	1 ⊢ ∃*𝑥{𝑥} = 𝐴

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃*wmo 2535  {csn 4578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-sn 4579

This theorem is referenced by:  pwfir  9215  euabsneu  47216