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Theorem eqsnd 4830
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 ((𝜑𝑥𝐴) → 𝑥 = 𝐵)
eqsnd.2 (𝜑𝐵𝐴)
Assertion
Ref Expression
eqsnd (𝜑𝐴 = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem eqsnd
StepHypRef Expression
1 eqsnd.1 . . 3 ((𝜑𝑥𝐴) → 𝑥 = 𝐵)
21ralrimiva 3146 . 2 (𝜑 → ∀𝑥𝐴 𝑥 = 𝐵)
3 eqsnd.2 . . . 4 (𝜑𝐵𝐴)
43ne0d 4342 . . 3 (𝜑𝐴 ≠ ∅)
5 eqsn 4829 . . 3 (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
64, 5syl 17 . 2 (𝜑 → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))
72, 6mpbird 257 1 (𝜑𝐴 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  c0 4333  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-sn 4627
This theorem is referenced by:  0ringidl  33449  lbsdiflsp0  33677  fiabv  42546
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