Theorem eqsnd 4785

Description: Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.) (Proof shortened by SN, 3-Jul-2025.)

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	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵)
2	1	ralrimiva 3153	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
3		eqsnd.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4	3	ne0d 4292	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
5		eqsn 4784	. . 3 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))
7	2, 6	mpbird 259	1 ⊢ (𝜑 → 𝐴 = {𝐵})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∅c0 4283  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-v 3455  df-dif 3905  df-ss 3919  df-nul 4284  df-sn 4580

This theorem is referenced by:  0ringidl  33567  dflring3  33653  dflring4  33654  lbsdiflsp0  33883  fiabv  43114  thinchom  50008