Theorem eqsnd 4855

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 3152	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
3		eqsnd.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4	3	ne0d 4365	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
5		eqsn 4854	. . 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 257	1 ⊢ (𝜑 → 𝐴 = {𝐵})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∅c0 4352  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-sn 4649

This theorem is referenced by:  0ringidl  33414  lbsdiflsp0  33639  fiabv  42491