Theorem snsspr1 4757

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 4118	. 2 ⊢ {𝐴} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4570	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3971	1 ⊢ {𝐴} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3887   ⊆ wss 3889  {csn 4567  {cpr 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-ss 3906  df-pr 4570

This theorem is referenced by:  snsstp1  4759  op1stb  5424  uniop  5469  1sdom2dom  9164  rankopb  9776  ltrelxr  11206  seqexw  13979  2strbas  17198  phlvsca  17313  prdshom  17430  ipobas  18497  ipolerval  18498  chnccat  18592  gsumpr  19930  lspprid1  20992  lsppratlem3  21147  lsppratlem4  21148  ex-dif  30493  ex-un  30494  ex-in  30495  idlsrgtset  33568  esplyind  33719  coinflippv  34628  pthhashvtx  35310  subfacp1lem2a  35362  altopthsn  36143  rankaltopb  36161  dvh3dim3N  41895  mapdindp2  42167  lspindp5  42216  algsca  43605  clsk1indlem2  44469  clsk1indlem3  44470  clsk1indlem1  44472  mnuprdlem4  44702  setc1onsubc  50077