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Theorem abvor0 3567
 Description: The class builder of a wff not containing the abstraction variable is either the universal class or the empty set. (Contributed by Mario Carneiro, 29-Aug-2013.)
Assertion
Ref Expression
abvor0 ({x φ} = V {x φ} = )
Distinct variable group:   φ,x

Proof of Theorem abvor0
StepHypRef Expression
1 id 19 . . . . . 6 (φφ)
2 vex 2862 . . . . . . 7 x V
32a1i 10 . . . . . 6 (φx V)
41, 32thd 231 . . . . 5 (φ → (φx V))
54abbi1dv 2469 . . . 4 (φ → {x φ} = V)
65con3i 127 . . 3 (¬ {x φ} = V → ¬ φ)
7 id 19 . . . . 5 φ → ¬ φ)
8 noel 3554 . . . . . 6 ¬ x
98a1i 10 . . . . 5 φ → ¬ x )
107, 92falsed 340 . . . 4 φ → (φx ))
1110abbi1dv 2469 . . 3 φ → {x φ} = )
126, 11syl 15 . 2 (¬ {x φ} = V → {x φ} = )
1312orri 365 1 ({x φ} = V {x φ} = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ∅c0 3550 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by:  abexv  4324
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