Theorem snsspr2 4616

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 4032	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4438	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtr4i 3888	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3821   ⊆ wss 3823  {csn 4435  {cpr 4437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-un 3828  df-in 3830  df-ss 3837  df-pr 4438

This theorem is referenced by:  snsstp2  4618  ord3ex  5134  ltrelxr  10496  2strop  16454  2strop1  16457  phlip  16508  prdsco  16591  ipotset  17619  gsumpr  18822  lsppratlem4  19638  ex-res  27992  subfacp1lem2a  32012  dvh3dim3N  38030  algvsca  39178  corclrcl  39415  mnuprdlem4  39986