Theorem snsspr2 4776

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 4125	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4587	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3980	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

This proof depends on syntax axioms:   ∪ cun 3897   ⊆ wss 3899  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-pr 4587

This theorem is used by:  snsstp2  4778  ord3ex  5349  ltrelxr  11363  2strop  17400  phlip  17515  prdsco  17632  ipotset  18700  gsumpr  20162  lsppratlem4  21421  ex-res  31035  esplyind  34200  subfacp1lem2a  35924  dvh3dim3N  42486  algvsca  44164  corclrcl  44692  mnuprdlem4  45244