Theorem prsstp12 3785

Description: A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Assertion

Ref	Expression
prsstp12	⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem prsstp12

Step	Hyp	Ref	Expression
1		ssun1 3335	. 2 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2		df-tp 3640	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sseqtrri 3227	1 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3163   ⊆ wss 3165  {csn 3632  {cpr 3633  {ctp 3634

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-tp 3640

This theorem is referenced by:  prsstp13  3786  prsstp23  3787  sstpr  3797