Theorem snsspr1 4740

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 4147	. 2 ⊢ {𝐴} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4563	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 4003	1 ⊢ {𝐴} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3933   ⊆ wss 3935  {csn 4560  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3940  df-in 3942  df-ss 3951  df-pr 4563

This theorem is referenced by:  snsstp1  4742  op1stb  5355  uniop  5397  rankopb  9275  ltrelxr  10696  seqexw  13379  2strbas  16597  2strbas1  16600  phlvsca  16651  prdshom  16734  ipobas  17759  ipolerval  17760  gsumpr  19069  lspprid1  19763  lsppratlem3  19915  lsppratlem4  19916  ex-dif  28196  ex-un  28197  ex-in  28198  coinflippv  31736  pthhashvtx  32369  subfacp1lem2a  32422  altopthsn  33417  rankaltopb  33435  dvh3dim3N  38579  mapdindp2  38851  lspindp5  38900  algsca  39774  clsk1indlem2  40385  clsk1indlem3  40386  clsk1indlem1  40388  mnuprdlem4  40604