Theorem snsspr2 4783

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 4132	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4594	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3987	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

This proof depends on syntax axioms:   ∪ cun 3904   ⊆ wss 3906  {csn 4591  {cpr 4593

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 2148  ax-9 2156  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923  df-pr 4594

This theorem is used by:  snsstp2  4785  ord3ex  5360  ltrelxr  11281  2strop  17306  phlip  17421  prdsco  17538  ipotset  18606  gsumpr  20048  lsppratlem4  21303  ex-res  30821  esplyind  33988  subfacp1lem2a  35685  dvh3dim3N  42256  algvsca  43938  corclrcl  44466  mnuprdlem4  45018