Theorem ab0orv 4309

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 Gino Giotto, 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 1918	. . 3 ⊢ Ⅎ𝑦𝜑
2		nf3 1790	. . 3 ⊢ (Ⅎ𝑦𝜑 ↔ (∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑))
3	1, 2	mpbi 229	. 2 ⊢ (∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑)
4		tbtru 1547	. . . . . 6 ⊢ (𝜑 ↔ (𝜑 ↔ ⊤))
5		df-clab 2716	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
6		sbv 2092	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	5, 6	bitr2i 275	. . . . . . 7 ⊢ (𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑})
8		tru 1543	. . . . . . . 8 ⊢ ⊤
9		vextru 2722	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
10	8, 9	2th 263	. . . . . . 7 ⊢ (⊤ ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
11	7, 10	bibi12i 339	. . . . . 6 ⊢ ((𝜑 ↔ ⊤) ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
12	4, 11	bitri 274	. . . . 5 ⊢ (𝜑 ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
13	12	albii 1823	. . . 4 ⊢ (∀𝑦𝜑 ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
14		dfcleq 2731	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊤}))
15		dfv2 3425	. . . . . 6 ⊢ V = {𝑥 ∣ ⊤}
16	15	eqcomi 2747	. . . . 5 ⊢ {𝑥 ∣ ⊤} = V
17	16	eqeq2i 2751	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊤} ↔ {𝑥 ∣ 𝜑} = V)
18	13, 14, 17	3bitr2i 298	. . 3 ⊢ (∀𝑦𝜑 ↔ {𝑥 ∣ 𝜑} = V)
19		equid 2016	. . . . . . 7 ⊢ 𝑦 = 𝑦
20	19	nbn3 373	. . . . . 6 ⊢ (¬ 𝜑 ↔ (𝜑 ↔ ¬ 𝑦 = 𝑦))
21		df-clab 2716	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∣ ¬ 𝑥 = 𝑥} ↔ [𝑦 / 𝑥] ¬ 𝑥 = 𝑥)
22		equid 2016	. . . . . . . . . . . 12 ⊢ 𝑥 = 𝑥
23	22, 19	2th 263	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑥 ↔ 𝑦 = 𝑦)
24	23	notbii 319	. . . . . . . . . 10 ⊢ (¬ 𝑥 = 𝑥 ↔ ¬ 𝑦 = 𝑦)
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = 𝑥 ↔ ¬ 𝑦 = 𝑦))
26	25	sbievw 2097	. . . . . . . 8 ⊢ ([𝑦 / 𝑥] ¬ 𝑥 = 𝑥 ↔ ¬ 𝑦 = 𝑦)
27	21, 26	bitr2i 275	. . . . . . 7 ⊢ (¬ 𝑦 = 𝑦 ↔ 𝑦 ∈ {𝑥 ∣ ¬ 𝑥 = 𝑥})
28	7, 27	bibi12i 339	. . . . . 6 ⊢ ((𝜑 ↔ ¬ 𝑦 = 𝑦) ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ¬ 𝑥 = 𝑥}))
29	20, 28	bitri 274	. . . . 5 ⊢ (¬ 𝜑 ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ¬ 𝑥 = 𝑥}))
30	29	albii 1823	. . . 4 ⊢ (∀𝑦 ¬ 𝜑 ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ¬ 𝑥 = 𝑥}))
31		dfcleq 2731	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ¬ 𝑥 = 𝑥} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ¬ 𝑥 = 𝑥}))
32		dfnul2 4256	. . . . . 6 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
33	32	eqcomi 2747	. . . . 5 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = ∅
34	33	eqeq2i 2751	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ¬ 𝑥 = 𝑥} ↔ {𝑥 ∣ 𝜑} = ∅)
35	30, 31, 34	3bitr2i 298	. . 3 ⊢ (∀𝑦 ¬ 𝜑 ↔ {𝑥 ∣ 𝜑} = ∅)
36	18, 35	orbi12i 911	. 2 ⊢ ((∀𝑦𝜑 ∨ ∀𝑦 ¬ 𝜑) ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))
37	3, 36	mpbi 229	1 ⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∨ wo 843  ∀wal 1537   = wceq 1539  ⊤wtru 1540  Ⅎwnf 1787  [wsb 2068   ∈ wcel 2108  {cab 2715  Vcvv 3422  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-v 3424  df-dif 3886  df-nul 4254

This theorem is referenced by: (None)