Theorem snsstp1 3859

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

Assertion

Ref	Expression
snsstp1	⊢ {A} ⊆ {A, B, C}

Proof of Theorem snsstp1

Step	Hyp	Ref	Expression
1		snsspr1 3857	. . 3 ⊢ {A} ⊆ {A, B}
2		ssun1 3427	. . 3 ⊢ {A, B} ⊆ ({A, B} ∪ {C})
3	1, 2	sstri 3282	. 2 ⊢ {A} ⊆ ({A, B} ∪ {C})
4		df-tp 3744	. 2 ⊢ {A, B, C} = ({A, B} ∪ {C})
5	3, 4	sseqtr4i 3305	1 ⊢ {A} ⊆ {A, B, C}

Syntax hints:   ∪ cun 3208   ⊆ wss 3258  {csn 3738  {cpr 3739  {ctp 3740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260  df-pr 3743  df-tp 3744

This theorem is referenced by: (None)