Theorem elsnc2g 3762

Description: There is only 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
elsnc2g	⊢ (B ∈ V → (A ∈ {B} ↔ A = B))

Proof of Theorem elsnc2g

Step	Hyp	Ref	Expression
1		elsni 3758	. 2 ⊢ (A ∈ {B} → A = B)
2		snidg 3759	. . 3 ⊢ (B ∈ V → B ∈ {B})
3		eleq1 2413	. . 3 ⊢ (A = B → (A ∈ {B} ↔ B ∈ {B}))
4	2, 3	syl5ibrcom 213	. 2 ⊢ (B ∈ V → (A = B → A ∈ {B}))
5	1, 4	impbid2 195	1 ⊢ (B ∈ V → (A ∈ {B} ↔ A = B))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sn 3742

This theorem is referenced by:  elsnc2  3763  fnfreclem2  6319  fnfreclem3  6320