Theorem snsspr1 4813

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 4168	. 2 ⊢ {𝐴} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4627	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 4015	1 ⊢ {𝐴} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3943   ⊆ wss 3945  {csn 4624  {cpr 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-un 3950  df-in 3952  df-ss 3962  df-pr 4627

This theorem is referenced by:  snsstp1  4815  op1stb  5467  uniop  5511  1sdom2dom  9265  rankopb  9869  ltrelxr  11299  seqexw  14008  2strbas  17196  2strbas1  17200  phlvsca  17324  prdshom  17442  ipobas  18516  ipolerval  18517  gsumpr  19903  lspprid1  20874  lsppratlem3  21030  lsppratlem4  21031  ex-dif  30226  ex-un  30227  ex-in  30228  idlsrgtset  33213  coinflippv  34097  pthhashvtx  34731  subfacp1lem2a  34784  altopthsn  35551  rankaltopb  35569  dvh3dim3N  40916  mapdindp2  41188  lspindp5  41237  algsca  42599  clsk1indlem2  43466  clsk1indlem3  43467  clsk1indlem1  43469  mnuprdlem4  43706