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Theorem elsn2g 4666
Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
elsn2g (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem elsn2g
StepHypRef Expression
1 elsni 4645 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2 snidg 4662 . . 3 (𝐵𝑉𝐵 ∈ {𝐵})
3 eleq1 2821 . . 3 (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵}))
42, 3syl5ibrcom 246 . 2 (𝐵𝑉 → (𝐴 = 𝐵𝐴 ∈ {𝐵}))
51, 4impbid2 225 1 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  {csn 4628
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-sn 4629
This theorem is referenced by:  elsn2  4667  mptiniseg  6238  elsuc2g  6433  extmptsuppeq  8172  fzosplitsni  13742  1nsgtrivd  19053  limcco  25409  ply1termlem  25716  mptprop  31915  bj-elsn12g  35936  elpmapat  38630  stirlinglem8  44787  dirkercncflem2  44810
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