Theorem elsn2g 4612

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 4588	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2		snidg 4608	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
3		eleq1 2819	. . 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 1541   ∈ wcel 2111  {csn 4571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sn 4572

This theorem is referenced by:  elsn2  4613  mptiniseg  6181  elsuc2g  6372  extmptsuppeq  8113  fzosplitsni  13674  1nsgtrivd  19081  limcco  25816  ply1termlem  26130  mptprop  32671  bj-elsn12g  37094  elpmapat  39803  stirlinglem8  46119  dirkercncflem2  46142  clnbgrel  47859