Theorem snsspr2 4786

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 4140	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4597	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3994	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3911   ⊆ wss 3913  {csn 4594  {cpr 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464  df-un 3918  df-ss 3930  df-pr 4597

This theorem is referenced by:  snsstp2  4788  ord3ex  5362  ltrelxr  11273  2strop  17292  phlip  17407  prdsco  17524  ipotset  18592  gsumpr  20028  lsppratlem4  21257  ex-res  30762  esplyind  33935  subfacp1lem2a  35630  dvh3dim3N  42173  algvsca  43857  corclrcl  44385  mnuprdlem4  44937