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Theorem prnzg 3836
 Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
prnzg (A V → {A, B} ≠ )

Proof of Theorem prnzg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 preq1 3799 . . 3 (x = A → {x, B} = {A, B})
21neeq1d 2529 . 2 (x = A → ({x, B} ≠ ↔ {A, B} ≠ ))
3 vex 2862 . . 3 x V
43prnz 3835 . 2 {x, B} ≠
52, 4vtoclg 2914 1 (A V → {A, B} ≠ )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∅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: (None)
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