Theorem eqsnd 30778

Description: Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.)

Hypotheses

Ref	Expression
eqsnd.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵)
eqsnd.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
eqsnd	⊢ (𝜑 → 𝐴 = {𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem eqsnd

Step	Hyp	Ref	Expression
1		eqsnd.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵)
2		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
3		eqsnd.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴)
5	2, 4	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
6	1, 5	impbida 797	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 = 𝐵))
7		velsn 4574	. . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
8	6, 7	bitr4di 288	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵}))
9	8	eqrdv 2736	1 ⊢ (𝜑 → 𝐴 = {𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-sn 4559

This theorem is referenced by:  0ringidl  31507  lbsdiflsp0  31609