Theorem ab0orv 4346

Description: The class abstraction defined by a formula not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) Reduce axiom usage. (Revised by GG, 30-Aug-2024.)

Assertion

Ref	Expression
ab0orv	⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)

Distinct variable group:   𝜑,𝑥

Proof of Theorem ab0orv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1941	. . 3 ⊢ Ⅎ𝑦𝜑
2		nf3 1813	. . 3 ⊢ (Ⅎ𝑦𝜑 ↔ (∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑))
3	1, 2	mpbi 233	. 2 ⊢ (∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑)
4		biidd 265	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
5	4	eqabcbw 2843	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝜑 ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
6		dfv2 3466	. . . . 5 ⊢ V = {𝑥 ∣ ⊤}
7	6	eqeq2i 2782	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = V ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
8		vextru 2754	. . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
9	8	tbt 372	. . . . 5 ⊢ (𝜑 ↔ (𝜑 ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
10	9	albii 1846	. . . 4 ⊢ (∀𝑦𝜑 ↔ ∀𝑦(𝜑 ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
11	5, 7, 10	3bitr4i 306	. . 3 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦𝜑)
12	4	ab0w 4342	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜑)
13	11, 12	orbi12i 927	. 2 ⊢ (({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) ↔ (∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑))
14	3, 13	mpbir 234	1 ⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∨ wo 860  ∀wal 1565   = wceq 1567  ⊤wtru 1568  Ⅎwnf 1810   ∈ wcel 2149  {cab 2747  Vcvv 3463  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-v 3465  df-dif 3916  df-nul 4295

This theorem is referenced by: (None)