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

Theorem snsspr1 3668
 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 3239 . 2 {𝐴} ⊆ ({𝐴} ∪ {𝐵})
2 df-pr 3534 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
31, 2sseqtrri 3132 1 {𝐴} ⊆ {𝐴, 𝐵}
 Colors of variables: wff set class Syntax hints:   ∪ cun 3069   ⊆ wss 3071  {csn 3527  {cpr 3528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pr 3534 This theorem is referenced by:  snsstp1  3670  ssprr  3683  uniop  4177  op1stb  4399  op1stbg  4400  ltrelxr  7832
 Copyright terms: Public domain W3C validator