Theorem snsspr2 4748

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 4107	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4564	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3958	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3885   ⊆ wss 3887  {csn 4561  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-pr 4564

This theorem is referenced by:  snsstp2  4750  ord3ex  5310  ltrelxr  11036  2strop  16936  2strop1  16940  phlip  17061  prdsco  17179  ipotset  18251  gsumpr  19556  lsppratlem4  20412  ex-res  28805  subfacp1lem2a  33142  dvh3dim3N  39463  algvsca  41007  corclrcl  41315  mnuprdlem4  41893