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 4180	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4591	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3962	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

This proof depends on syntax axioms:   ⊆ wss 3900   ∪ cun 4152  {csn 4588  {cpr 4590

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-pr 4591

This theorem is used by:  snsstp2  4778  ord3ex  5356  ltrelxr  11409  2strop  17451  phlip  17568  prdsco  17689  ipotset  18757  gsumpr  20220  lsppratlem4  21479  cnfldeuf  21743  ex-res  31099  esplyind  34318  subfacp1lem2a  36153  dvh3dim3N  42772  algvsca  44441  corclrcl  44968  mnuprdlem4  45520