MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snsspr2 Structured version   Visualization version   GIF version

Theorem snsspr2 4491
Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
Assertion
Ref Expression
snsspr2 {𝐵} ⊆ {𝐴, 𝐵}

Proof of Theorem snsspr2
StepHypRef Expression
1 ssun2 3920 . 2 {𝐵} ⊆ ({𝐴} ∪ {𝐵})
2 df-pr 4324 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
31, 2sseqtr4i 3779 1 {𝐵} ⊆ {𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:  cun 3713  wss 3715  {csn 4321  {cpr 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-in 3722  df-ss 3729  df-pr 4324
This theorem is referenced by:  snsstp2  4493  ord3ex  5005  ltrelxr  10291  2strop  16187  2strop1  16190  phlip  16241  prdsco  16330  ipotset  17358  lsppratlem4  19352  ex-res  27609  subfacp1lem2a  31469  dvh3dim3N  37240  algvsca  38254  corclrcl  38501  gsumpr  42649
  Copyright terms: Public domain W3C validator