Theorem elsni 3757

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 3755	. 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 3737

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-sn 3741

This theorem is referenced by:  elsnc2g  3761  disjsn2  3787  sssn  3864  unsneqsn  3887  fvconst  5440  fvunsn  5444  xpnedisj  5513  enadjlem1  6059  enadj  6060