Theorem prsstp12 3538

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 3135	. 2 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2		df-tp 3406	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sseqtr4i 3032	1 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 2971   ⊆ wss 2973  {csn 3398  {cpr 3399  {ctp 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-tp 3406

This theorem is referenced by:  prsstp13  3539  prsstp23  3540  sstpr  3549