Theorem snsstp2 3824

Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Assertion

Ref	Expression
snsstp2	⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem snsstp2

Step	Hyp	Ref	Expression
1		snsspr2 3822	. . 3 ⊢ {𝐵} ⊆ {𝐴, 𝐵}
2		ssun1 3370	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3236	. 2 ⊢ {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 3677	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 3262	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3198   ⊆ wss 3200  {csn 3669  {cpr 3670  {ctp 3671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pr 3676  df-tp 3677

This theorem is referenced by:  prdsplusg  13359  mpocnfldadd  14574  cnfldle  14580  psrplusgg  14691