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Theorem eqsnd 30296
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 488 . . . . 5 ((𝜑𝑥 = 𝐵) → 𝑥 = 𝐵)
3 eqsnd.2 . . . . . 6 (𝜑𝐵𝐴)
43adantr 484 . . . . 5 ((𝜑𝑥 = 𝐵) → 𝐵𝐴)
52, 4eqeltrd 2914 . . . 4 ((𝜑𝑥 = 𝐵) → 𝑥𝐴)
61, 5impbida 800 . . 3 (𝜑 → (𝑥𝐴𝑥 = 𝐵))
7 velsn 4555 . . 3 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
86, 7syl6bbr 292 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝐵}))
98eqrdv 2820 1 (𝜑𝐴 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {csn 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-sn 4540
This theorem is referenced by:  lbsdiflsp0  31079
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