Theorem ab0orv 4337

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 1916	. . 3 ⊢ Ⅎ𝑦𝜑
2		nf3 1788	. . 3 ⊢ (Ⅎ𝑦𝜑 ↔ (∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑))
3	1, 2	mpbi 230	. 2 ⊢ (∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑)
4		biidd 262	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
5	4	eqabcbw 2811	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝜑 ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
6		dfv2 3445	. . . . 5 ⊢ V = {𝑥 ∣ ⊤}
7	6	eqeq2i 2750	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = V ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤})
8		vextru 2722	. . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
9	8	tbt 369	. . . . 5 ⊢ (𝜑 ↔ (𝜑 ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
10	9	albii 1821	. . . 4 ⊢ (∀𝑦𝜑 ↔ ∀𝑦(𝜑 ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
11	5, 7, 10	3bitr4i 303	. . 3 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦𝜑)
12	4	ab0w 4333	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜑)
13	11, 12	orbi12i 915	. 2 ⊢ (({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) ↔ (∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑))
14	3, 13	mpbir 231	1 ⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 848  ∀wal 1540   = wceq 1542  ⊤wtru 1543  Ⅎwnf 1785   ∈ wcel 2114  {cab 2715  Vcvv 3442  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-v 3444  df-dif 3906  df-nul 4288

This theorem is referenced by: (None)