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Theorem elsn2g 4157
 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 4142 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2 snidg 4153 . . 3 (𝐵𝑉𝐵 ∈ {𝐵})
3 eleq1 2676 . . 3 (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵}))
42, 3syl5ibrcom 236 . 2 (𝐵𝑉 → (𝐴 = 𝐵𝐴 ∈ {𝐵}))
51, 4impbid2 215 1 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sn 4126 This theorem is referenced by:  elsn2  4158  mptiniseg  5546  elsuc2g  5710  extmptsuppeq  7206  fzosplitsni  12444  limcco  23463  ply1termlem  23763  elpmapat  34068  stirlinglem8  38974  dirkercncflem2  38997
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