Theorem snsspr2 4745

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 4103	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4561	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3954	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3881   ⊆ wss 3883  {csn 4558  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-pr 4561

This theorem is referenced by:  snsstp2  4747  ord3ex  5305  ltrelxr  10967  2strop  16862  2strop1  16866  phlip  16986  prdsco  17096  ipotset  18166  gsumpr  19471  lsppratlem4  20327  ex-res  28706  subfacp1lem2a  33042  dvh3dim3N  39390  algvsca  40923  corclrcl  41204  mnuprdlem4  41782