Theorem snsstp3 4349

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

Assertion

Ref	Expression
snsstp3	⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem snsstp3

Step	Hyp	Ref	Expression
1		ssun2 3777	. 2 ⊢ {𝐶} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2		df-tp 4182	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sseqtr4i 3638	1 ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3572   ⊆ wss 3574  {csn 4177  {cpr 4179  {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-tp 4182

This theorem is referenced by:  fr3nr  6979  rngmulr  16003  srngmulr  16011  lmodsca  16020  ipsmulr  16027  ipsip  16030  phlsca  16037  topgrptset  16045  otpsle  16054  otpsleOLD  16058  odrngmulr  16069  odrngds  16072  prdsmulr  16119  prdsip  16121  prdsds  16124  imasds  16173  imasmulr  16178  imasip  16181  fuccofval  16619  setccofval  16732  catccofval  16750  estrccofval  16769  xpccofval  16822  psrmulr  19384  cnfldmul  19752  cnfldds  19756  trkgitv  25346  signswch  30638  algmulr  37750  clsk1indlem1  38343  rngccofvalALTV  41987  ringccofvalALTV  42050