Theorem abvor0 3568

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

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 ⊢ (φ → φ)
2		vex 2863	. . . . . . 7 ⊢ x ∈ V
3	2	a1i 10	. . . . . 6 ⊢ (φ → x ∈ V)
4	1, 3	2thd 231	. . . . 5 ⊢ (φ → (φ ↔ x ∈ V))
5	4	eqabcdv 2470	. . . 4 ⊢ (φ → {x ∣ φ} = V)
6	5	con3i 127	. . 3 ⊢ (¬ {x ∣ φ} = V → ¬ φ)
7		id 19	. . . . 5 ⊢ (¬ φ → ¬ φ)
8		noel 3555	. . . . . 6 ⊢ ¬ x ∈ ∅
9	8	a1i 10	. . . . 5 ⊢ (¬ φ → ¬ x ∈ ∅)
10	7, 9	2falsed 340	. . . 4 ⊢ (¬ φ → (φ ↔ x ∈ ∅))
11	10	eqabcdv 2470	. . 3 ⊢ (¬ φ → {x ∣ φ} = ∅)
12	6, 11	syl 15	. 2 ⊢ (¬ {x ∣ φ} = V → {x ∣ φ} = ∅)
13	12	orri 365	1 ⊢ ({x ∣ φ} = V ∨ {x ∣ φ} = ∅)

Syntax hints:  ¬ wn 3   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  abexv  4325