Theorem snsspr1 4763

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 4125	. 2 ⊢ {𝐴} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4576	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3979	1 ⊢ {𝐴} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3895   ⊆ wss 3897  {csn 4573  {cpr 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914  df-pr 4576

This theorem is referenced by:  snsstp1  4765  op1stb  5409  uniop  5453  1sdom2dom  9138  rankopb  9745  ltrelxr  11173  seqexw  13924  2strbas  17139  phlvsca  17254  prdshom  17371  ipobas  18437  ipolerval  18438  chnccat  18532  gsumpr  19867  lspprid1  20930  lsppratlem3  21086  lsppratlem4  21087  ex-dif  30403  ex-un  30404  ex-in  30405  idlsrgtset  33473  coinflippv  34497  pthhashvtx  35172  subfacp1lem2a  35224  altopthsn  36003  rankaltopb  36021  dvh3dim3N  41496  mapdindp2  41768  lspindp5  41817  algsca  43218  clsk1indlem2  44083  clsk1indlem3  44084  clsk1indlem1  44086  mnuprdlem4  44316  setc1onsubc  49642