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Theorem eqsnd 29939
 Description: Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.)
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 . . . 4 ((𝜑𝑥𝐴) → 𝑥 = 𝐵)
2 simpr 479 . . . . 5 ((𝜑𝑥 = 𝐵) → 𝑥 = 𝐵)
3 eqsnd.2 . . . . . 6 (𝜑𝐵𝐴)
43adantr 474 . . . . 5 ((𝜑𝑥 = 𝐵) → 𝐵𝐴)
52, 4eqeltrd 2859 . . . 4 ((𝜑𝑥 = 𝐵) → 𝑥𝐴)
61, 5impbida 791 . . 3 (𝜑 → (𝑥𝐴𝑥 = 𝐵))
7 vex 3401 . . . 4 𝑥 ∈ V
87elsn 4413 . . 3 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
96, 8syl6bbr 281 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝐵}))
109eqrdv 2776 1 (𝜑𝐴 = {𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {csn 4398 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-sn 4399 This theorem is referenced by:  lbsdiflsp0  30448
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