New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  prnz GIF version

Theorem prnz 3835
 Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
Hypothesis
Ref Expression
prnz.1 A V
Assertion
Ref Expression
prnz {A, B} ≠

Proof of Theorem prnz
StepHypRef Expression
1 prnz.1 . . 3 A V
21prid1 3827 . 2 A {A, B}
3 ne0i 3556 . 2 (A {A, B} → {A, B} ≠ )
42, 3ax-mp 8 1 {A, B} ≠
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859  ∅c0 3550  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-nul 3551  df-sn 3741  df-pr 3742 This theorem is referenced by:  prnzg  3836
 Copyright terms: Public domain W3C validator