Theorem snsspr2 4768

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 4128	. 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2		df-pr 4580	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	1, 2	sseqtrri 3980	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵}

Syntax hints:   ∪ cun 3896   ⊆ wss 3898  {csn 4577  {cpr 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903  df-ss 3915  df-pr 4580

This theorem is referenced by:  snsstp2  4770  ord3ex  5329  ltrelxr  11184  2strop  17147  phlip  17262  prdsco  17379  ipotset  18447  gsumpr  19875  lsppratlem4  21096  ex-res  30442  esplyind  33659  subfacp1lem2a  35296  dvh3dim3N  41621  algvsca  43335  corclrcl  43864  mnuprdlem4  44432