Theorem eqimssi 3304

Description: Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)

Hypothesis

Ref	Expression
eqimssi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
eqimssi	⊢ 𝐴 ⊆ 𝐵

Proof of Theorem eqimssi

Step	Hyp	Ref	Expression
1		ssid 3268	. 2 ⊢ 𝐴 ⊆ 𝐴
2		eqimssi.1	. 2 ⊢ 𝐴 = 𝐵
3	1, 2	sseqtri 3282	1 ⊢ 𝐴 ⊆ 𝐵

This proof depends on syntax axioms:   = wceq 1402   ⊆ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  funi  5409  fpr  5897  elfzo1  10605  sumsplitdc  12201  isumlessdc  12265  nconstwlpolem0  17125