Theorem snsspr2 4770

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 4177	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4587	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3957	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

This proof depends on syntax axioms:   ⊆ wss 3895   ∪ cun 4149  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-un 4151  df-pr 4587

This theorem is used by:  snsstp2  4772  ord3ex  5348  ltrelxr  11374  2strop  17413  phlip  17528  prdsco  17645  ipotset  18713  gsumpr  20175  lsppratlem4  21434  ex-res  31048  esplyind  34254  subfacp1lem2a  36032  dvh3dim3N  42600  algvsca  44274  corclrcl  44801  mnuprdlem4  45353