Theorem mosneq 4775

Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3700. (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 2845	. . . 4 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦})
2		vex 3499	. . . . 5 ⊢ 𝑥 ∈ V
3	2	sneqr 4773	. . . 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 4579	. . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
7	6	eqeq1d 2825	. . 3 ⊢ (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴))
8	7	mo4 2650	. 2 ⊢ (∃*𝑥{𝑥} = 𝐴 ↔ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦))
9	5, 8	mpbir 233	1 ⊢ ∃*𝑥{𝑥} = 𝐴

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃*wmo 2620  {csn 4569

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-sn 4570

This theorem is referenced by:  euabsneu  43270