Theorem iresn0n0 6083

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 5573	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
2		opabresid 6079	. . 3 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	2	eqeq1i 2745	. 2 ⊢ (( I ↾ 𝐴) = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅)
4		nel02 4362	. . . . 5 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
5	4	intnanrd 489	. . . 4 ⊢ (𝐴 = ∅ → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
6	5	alrimivv 1927	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
7		ianor 982	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥))
8	7	albii 1817	. . . . . 6 ⊢ (∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ ∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥))
9		19.32v 1939	. . . . . . 7 ⊢ (∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥))
10		id 22	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐴)
11		ax6v 1968	. . . . . . . . 9 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥
12	11	pm2.21i 119	. . . . . . . 8 ⊢ (∀𝑦 ¬ 𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐴)
13	10, 12	jaoi 856	. . . . . . 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 1809	. . . 4 ⊢ (∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
17		eq0 4373	. . . 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 846  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∅c0 4352  {copab 5228   I cid 5592   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-res 5712

This theorem is referenced by: (None)