Theorem sneqbg 3851

Description: Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
sneqbg	⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem sneqbg

Step	Hyp	Ref	Expression
1		sneqrg 3850	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2		sneq 3684	. 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
3	1, 2	impbid1 142	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2202  {csn 3673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sn 3679

This theorem is referenced by:  suppval1  6417  suppsnopdc  6428  eqsndc  7138  infpwfidom  7452  s111  11257