Theorem ralf0 4489

Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) Avoid df-clel 2809, ax-8 2110. (Revised by GG, 2-Sep-2024.)

Hypothesis

Ref	Expression
ralf0.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
ralf0	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralf0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26 3098	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 ∧ 𝜑) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑))
2		pm2.24 124	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝜑 → ⊥))
3	2	impcom 407	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ 𝜑) → ⊥)
4	3	ralimi 3073	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 ∧ 𝜑) → ∀𝑥 ∈ 𝐴 ⊥)
5		df-ral 3052	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ⊥))
6		dfnot 1559	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 → ⊥))
7	6	bicomi 224	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → ⊥) ↔ ¬ 𝑥 ∈ 𝐴)
8	7	albii 1819	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ⊥) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
9	5, 8	sylbb 219	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ⊥ → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
10		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
11		falim 1557	. . . . . . . . . . 11 ⊢ (⊥ → 𝑥 ∈ 𝐴)
12	10, 11	pm5.21ni 377	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ ⊥))
13		df-clab 2714	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
14		sbv 2088	. . . . . . . . . . 11 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥)
15	13, 14	bitri 275	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
16	12, 15	bitr4di 289	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
17	16	alimi 1811	. . . . . . . 8 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
18		dfcleq 2728	. . . . . . . 8 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
19	17, 18	sylibr 234	. . . . . . 7 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 → 𝐴 = {𝑦 ∣ ⊥})
20		dfnul4 4310	. . . . . . 7 ⊢ ∅ = {𝑦 ∣ ⊥}
21	19, 20	eqtr4di 2788	. . . . . 6 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 → 𝐴 = ∅)
22	4, 9, 21	3syl 18	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 ∧ 𝜑) → 𝐴 = ∅)
23	1, 22	sylbir 235	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)
24	23	ex 412	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = ∅))
25		ralf0.1	. . . 4 ⊢ ¬ 𝜑
26	25	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑)
27	24, 26	mprg 3057	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = ∅)
28		rzal 4484	. 2 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
29	27, 28	impbii 209	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ⊥wfal 1552  [wsb 2064   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∅c0 4308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-ral 3052  df-dif 3929  df-nul 4309

This theorem is referenced by:  rext0  44963