Theorem snsspr2 4728

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 4087	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4544	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3938	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3864   ⊆ wss 3866  {csn 4541  {cpr 4543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-un 3871  df-in 3873  df-ss 3883  df-pr 4544

This theorem is referenced by:  snsstp2  4730  ord3ex  5280  ltrelxr  10894  2strop  16780  2strop1  16783  phlip  16884  prdsco  16973  ipotset  18039  gsumpr  19340  lsppratlem4  20187  ex-res  28524  subfacp1lem2a  32855  dvh3dim3N  39200  algvsca  40710  corclrcl  40992  mnuprdlem4  41566