Theorem snsspr2 4775

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 4138	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4588	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3993	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3909   ⊆ wss 3911  {csn 4585  {cpr 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-un 3916  df-ss 3928  df-pr 4588

This theorem is referenced by:  snsstp2  4777  ord3ex  5337  ltrelxr  11211  2strop  17175  phlip  17290  prdsco  17407  ipotset  18468  gsumpr  19861  lsppratlem4  21036  ex-res  30343  subfacp1lem2a  35140  dvh3dim3N  41416  algvsca  43140  corclrcl  43669  mnuprdlem4  44237