Theorem snsspr2 4286

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 3739	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4128	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtr4i 3601	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3538   ⊆ wss 3540  {csn 4125  {cpr 4127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-pr 4128

This theorem is referenced by:  snsstp2  4288  ord3ex  4782  ltrelxr  9978  2strop  15811  2strop1  15814  phlip  15862  prdsco  15951  ipotset  16980  lsppratlem4  18971  constr3pthlem1  26183  ex-res  26690  subfacp1lem2a  30416  dvh3dim3N  35756  algvsca  36771  corclrcl  37018  gsumpr  41932