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Theorem elsn2g 4566
 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 4545 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2 snidg 4562 . . 3 (𝐵𝑉𝐵 ∈ {𝐵})
3 eleq1 2877 . . 3 (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵}))
42, 3syl5ibrcom 250 . 2 (𝐵𝑉 → (𝐴 = 𝐵𝐴 ∈ {𝐵}))
51, 4impbid2 229 1 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {csn 4528 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 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-sn 4529 This theorem is referenced by:  elsn2  4567  mptiniseg  6065  elsuc2g  6234  extmptsuppeq  7853  fzosplitsni  13163  1nsgtrivd  18339  limcco  24537  ply1termlem  24844  mptprop  30502  bj-elsn12g  34628  elpmapat  37211  stirlinglem8  42891  dirkercncflem2  42914
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