Theorem iresn0n0 6021

Description: The identity function restricted to a class 𝐴 is empty iff the class 𝐴 is empty. (Contributed by AV, 30-Jan-2024.)

Assertion

Ref	Expression
iresn0n0	⊢ (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)

Proof of Theorem iresn0n0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opab0 5510	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
2		opabresid 6017	. . 3 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	2	eqeq1i 2742	. 2 ⊢ (( I ↾ 𝐴) = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅)
4		nel02 4293	. . . . 5 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
5	4	intnanrd 489	. . . 4 ⊢ (𝐴 = ∅ → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
6	5	alrimivv 1930	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
7		ianor 984	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥))
8	7	albii 1821	. . . . . 6 ⊢ (∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ ∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥))
9		19.32v 1942	. . . . . . 7 ⊢ (∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥))
10		id 22	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐴)
11		ax6v 1970	. . . . . . . . 9 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥
12	11	pm2.21i 119	. . . . . . . 8 ⊢ (∀𝑦 ¬ 𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐴)
13	10, 12	jaoi 858	. . . . . . 7 ⊢ ((¬ 𝑥 ∈ 𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥) → ¬ 𝑥 ∈ 𝐴)
14	9, 13	sylbi 217	. . . . . 6 ⊢ (∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥) → ¬ 𝑥 ∈ 𝐴)
15	8, 14	sylbi 217	. . . . 5 ⊢ (∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → ¬ 𝑥 ∈ 𝐴)
16	15	alimi 1813	. . . 4 ⊢ (∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
17		eq0 4304	. . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
18	16, 17	sylibr 234	. . 3 ⊢ (∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = ∅)
19	6, 18	impbii 209	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
20	1, 3, 19	3bitr4ri 304	1 ⊢ (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 848  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∅c0 4287  {copab 5162   I cid 5526   ↾ cres 5634

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-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-res 5644

This theorem is referenced by: (None)