Theorem elsn2g 4669

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 4648	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2		snidg 4665	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
3		eleq1 2827	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵}))
4	2, 3	syl5ibrcom 247	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵}))
5	1, 4	impbid2 226	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sn 4632

This theorem is referenced by:  elsn2  4670  mptiniseg  6261  elsuc2g  6455  extmptsuppeq  8212  fzosplitsni  13814  1nsgtrivd  19205  limcco  25943  ply1termlem  26257  mptprop  32713  bj-elsn12g  37043  elpmapat  39747  stirlinglem8  46037  dirkercncflem2  46060  clnbgrel  47753