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	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem elsn2g

Step	Hyp	Ref	Expression
1		elsni 4558	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2		snidg 4575	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
3		eleq1 2825	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵}))
4	2, 3	syl5ibrcom 250	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵}))
5	1, 4	impbid2 229	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

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