Theorem snsspr2 4818

Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)

Assertion

Ref	Expression
snsspr2	⊢ {𝐵} ⊆ {𝐴, 𝐵}

Proof of Theorem snsspr2

Step	Hyp	Ref	Expression
1		ssun2 4173	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4631	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 4019	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3946   ⊆ wss 3948  {csn 4628  {cpr 4630

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-pr 4631

This theorem is referenced by:  snsstp2  4820  ord3ex  5385  ltrelxr  11279  2strop  17172  2strop1  17176  phlip  17300  prdsco  17418  ipotset  18490  gsumpr  19864  lsppratlem4  20908  ex-res  29949  subfacp1lem2a  34457  dvh3dim3N  40623  algvsca  42226  corclrcl  42760  mnuprdlem4  43336