Theorem snsspr1 3755

Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)

Assertion

Ref	Expression
snsspr1	⊢ {𝐴} ⊆ {𝐴, 𝐵}

Proof of Theorem snsspr1

Step	Hyp	Ref	Expression
1		ssun1 3313	. 2 ⊢ {𝐴} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 3614	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3205	1 ⊢ {𝐴} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3142   ⊆ wss 3144  {csn 3607  {cpr 3608

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pr 3614

This theorem is referenced by:  snsstp1  3757  ssprr  3771  uniop  4270  op1stb  4493  op1stbg  4494  ltrelxr  8036  lspprid1  13688