Theorem snsspr2 4767

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 4129	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4579	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3984	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3900   ⊆ wss 3902  {csn 4576  {cpr 4578

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 3907  df-ss 3919  df-pr 4579

This theorem is referenced by:  snsstp2  4769  ord3ex  5325  ltrelxr  11173  2strop  17140  phlip  17255  prdsco  17372  ipotset  18439  gsumpr  19868  lsppratlem4  21088  ex-res  30419  subfacp1lem2a  35222  dvh3dim3N  41494  algvsca  43217  corclrcl  43746  mnuprdlem4  44314