MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsn2g Structured version   Visualization version   GIF version

Theorem elsn2g 4579
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 4558 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2 snidg 4575 . . 3 (𝐵𝑉𝐵 ∈ {𝐵})
3 eleq1 2825 . . 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 1543  wcel 2110  {csn 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-sn 4542
This theorem is referenced by:  elsn2  4580  mptiniseg  6102  elsuc2g  6281  extmptsuppeq  7930  fzosplitsni  13353  1nsgtrivd  18590  limcco  24790  ply1termlem  25097  mptprop  30751  bj-elsn12g  34968  elpmapat  37515  stirlinglem8  43297  dirkercncflem2  43320
  Copyright terms: Public domain W3C validator