Theorem snsspr2 4741

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 4148	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4563	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 4003	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3933   ⊆ wss 3935  {csn 4560  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3940  df-in 3942  df-ss 3951  df-pr 4563

This theorem is referenced by:  snsstp2  4743  ord3ex  5279  ltrelxr  10696  2strop  16598  2strop1  16601  phlip  16652  prdsco  16735  ipotset  17761  gsumpr  19069  lsppratlem4  19916  ex-res  28214  subfacp1lem2a  32422  dvh3dim3N  38579  algvsca  39775  corclrcl  40045  mnuprdlem4  40604