Theorem elsni 3758

Description: There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)

Assertion

Ref	Expression
elsni	⊢ (A ∈ {B} → A = B)

Proof of Theorem elsni

Step	Hyp	Ref	Expression
1		elsncg 3756	. 2 ⊢ (A ∈ {B} → (A ∈ {B} ↔ A = B))
2	1	ibi 232	1 ⊢ (A ∈ {B} → A = B)

Syntax hints:   → wi 4   = 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:  elsnc2g  3762  disjsn2  3788  sssn  3865  unsneqsn  3888  fvconst  5441  fvunsn  5445  xpnedisj  5514  enadjlem1  6060  enadj  6061