MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snsspr1 Structured version   Visualization version   GIF version

Theorem snsspr1 4377
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 3809 . 2 {𝐴} ⊆ ({𝐴} ∪ {𝐵})
2 df-pr 4213 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
31, 2sseqtr4i 3671 1 {𝐴} ⊆ {𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:  cun 3605  wss 3607  {csn 4210  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-pr 4213
This theorem is referenced by:  snsstp1  4379  op1stb  4969  uniop  5006  rankopb  8753  ltrelxr  10137  2strbas  16031  2strbas1  16034  phlvsca  16085  prdshom  16174  ipobas  17202  ipolerval  17203  lspprid1  19045  lsppratlem3  19197  lsppratlem4  19198  ex-dif  27410  ex-un  27411  ex-in  27412  coinflippv  30673  subfacp1lem2a  31288  altopthsn  32193  rankaltopb  32211  dvh3dim3N  37055  mapdindp2  37327  lspindp5  37376  algsca  38068  clsk1indlem2  38657  clsk1indlem3  38658  clsk1indlem1  38660  gsumpr  42464
  Copyright terms: Public domain W3C validator