Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  snsspr1 GIF version

Theorem snsspr1 3539
 Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
snsspr1 {𝐴} ⊆ {𝐴, 𝐵}

Proof of Theorem snsspr1
StepHypRef Expression
1 ssun1 3133 . 2 {𝐴} ⊆ ({𝐴} ∪ {𝐵})
2 df-pr 3409 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
31, 2sseqtr4i 3005 1 {𝐴} ⊆ {𝐴, 𝐵}
 Colors of variables: wff set class Syntax hints:   ∪ cun 2942   ⊆ wss 2944  {csn 3402  {cpr 3403 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pr 3409 This theorem is referenced by:  snsstp1  3541  ssprr  3554  uniop  4019  op1stb  4236  op1stbg  4237  ltrelxr  7138
 Copyright terms: Public domain W3C validator