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Theorem elsn2g 4664
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 4643 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2 snidg 4660 . . 3 (𝐵𝑉𝐵 ∈ {𝐵})
3 eleq1 2829 . . 3 (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵}))
42, 3syl5ibrcom 247 . 2 (𝐵𝑉 → (𝐴 = 𝐵𝐴 ∈ {𝐵}))
51, 4impbid2 226 1 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sn 4627
This theorem is referenced by:  elsn2  4665  mptiniseg  6259  elsuc2g  6453  extmptsuppeq  8213  fzosplitsni  13817  1nsgtrivd  19192  limcco  25928  ply1termlem  26242  mptprop  32707  bj-elsn12g  37061  elpmapat  39766  stirlinglem8  46096  dirkercncflem2  46119  clnbgrel  47815
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