Theorem sneqi 3644

Description: Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)

Hypothesis

Ref	Expression
sneqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
sneqi	⊢ {𝐴} = {𝐵}

Proof of Theorem sneqi

Step	Hyp	Ref	Expression
1		sneqi.1	. 2 ⊢ 𝐴 = 𝐵
2		sneq 3643	. 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
3	1, 2	ax-mp 5	1 ⊢ {𝐴} = {𝐵}

Syntax hints:   = wceq 1372  {csn 3632

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-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-sn 3638

This theorem is referenced by:  funopsn  5761  fnressn  5769  fressnfv  5770  snriota  5928  xpassen  6924  ennnfonelem1  12749  strle1g  12909  imasplusg  13111  ghmeqker  13578