ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snsstp1 GIF version

Theorem snsstp1 3754
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
StepHypRef Expression
1 snsspr1 3752 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
2 ssun1 3310 . . 3 {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
31, 2sstri 3176 . 2 {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 3612 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
53, 4sseqtrri 3202 1 {𝐴} ⊆ {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  cun 3139  wss 3141  {csn 3604  {cpr 3605  {ctp 3606
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pr 3611  df-tp 3612
This theorem is referenced by:  cnfldbas  13672
  Copyright terms: Public domain W3C validator