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Theorem eqsnd 4835
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 3144 . 2 (𝜑 → ∀𝑥𝐴 𝑥 = 𝐵)
3 eqsnd.2 . . . 4 (𝜑𝐵𝐴)
43ne0d 4348 . . 3 (𝜑𝐴 ≠ ∅)
5 eqsn 4834 . . 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 1537  wcel 2106  wne 2938  wral 3059  c0 4339  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-sn 4632
This theorem is referenced by:  0ringidl  33429  lbsdiflsp0  33654  fiabv  42523
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