Theorem eqsnd 4797

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 3163	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
3		eqsnd.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4	3	ne0d 4303	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
5		eqsn 4796	. . 3 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))
7	2, 6	mpbird 260	1 ⊢ (𝜑 → 𝐴 = {𝐵})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∅c0 4294  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-v 3465  df-dif 3916  df-ss 3930  df-nul 4295  df-sn 4592

This theorem is referenced by:  0ringidl  21334  tglineinsn  28875  dflring3  33728  dflring4  33729  lbsdiflsp0  33957  fiabv  43189  thinchom  50083