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Theorem eqsnd 31260
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 486 . . . . 5 ((𝜑𝑥 = 𝐵) → 𝑥 = 𝐵)
3 eqsnd.2 . . . . . 6 (𝜑𝐵𝐴)
43adantr 482 . . . . 5 ((𝜑𝑥 = 𝐵) → 𝐵𝐴)
52, 4eqeltrd 2839 . . . 4 ((𝜑𝑥 = 𝐵) → 𝑥𝐴)
61, 5impbida 800 . . 3 (𝜑 → (𝑥𝐴𝑥 = 𝐵))
7 velsn 4601 . . 3 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
86, 7bitr4di 289 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝐵}))
98eqrdv 2736 1 (𝜑𝐴 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {csn 4585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-sn 4586
This theorem is referenced by:  0ringidl  31998  lbsdiflsp0  32111
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