Theorem mosneq 4774

Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3648. (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 2761	. . . 4 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦})
2		vex 3435	. . . . 5 ⊢ 𝑥 ∈ V
3	2	sneqr 4772	. . . 4 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
4	1, 3	syl 17	. . 3 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
5	4	gen2 1803	. 2 ⊢ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)
6		sneq 4566	. . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
7	6	eqeq1d 2741	. . 3 ⊢ (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴))
8	7	mo4 2570	. 2 ⊢ (∃*𝑥{𝑥} = 𝐴 ↔ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦))
9	5, 8	mpbir 232	1 ⊢ ∃*𝑥{𝑥} = 𝐴

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃*wmo 2541  {csn 4556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-sn 4557

This theorem is referenced by:  pwfir  9218  euabsneu  47499