Theorem iresn0n0 6046

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 5547	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
2		opabresid 6042	. . 3 ⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
3	2	eqeq1i 2731	. 2 ⊢ (( I ↾ 𝐴) = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅)
4		nel02 4327	. . . . 5 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
5	4	intnanrd 489	. . . 4 ⊢ (𝐴 = ∅ → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
6	5	alrimivv 1923	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
7		ianor 978	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥))
8	7	albii 1813	. . . . . 6 ⊢ (∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ ∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥))
9		19.32v 1935	. . . . . . 7 ⊢ (∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥))
10		id 22	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐴)
11		ax6v 1964	. . . . . . . . 9 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥
12	11	pm2.21i 119	. . . . . . . 8 ⊢ (∀𝑦 ¬ 𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐴)
13	10, 12	jaoi 854	. . . . . . 7 ⊢ ((¬ 𝑥 ∈ 𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥) → ¬ 𝑥 ∈ 𝐴)
14	9, 13	sylbi 216	. . . . . 6 ⊢ (∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥) → ¬ 𝑥 ∈ 𝐴)
15	8, 14	sylbi 216	. . . . 5 ⊢ (∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → ¬ 𝑥 ∈ 𝐴)
16	15	alimi 1805	. . . 4 ⊢ (∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
17		eq0 4338	. . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
18	16, 17	sylibr 233	. . 3 ⊢ (∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = ∅)
19	6, 18	impbii 208	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥))
20	1, 3, 19	3bitr4ri 304	1 ⊢ (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 844  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∅c0 4317  {copab 5203   I cid 5566   ↾ cres 5671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-res 5681

This theorem is referenced by: (None)