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Mirrors > Home > MPE Home > Th. List > iresn0n0 | Structured version Visualization version GIF version |
Description: The identity function restricted to a class 𝐴 is empty iff the class 𝐴 is empty. (Contributed by AV, 30-Jan-2024.) |
Ref | Expression |
---|---|
iresn0n0 | ⊢ (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opab0 5564 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)) | |
2 | opabresid 6070 | . . 3 ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
3 | 2 | eqeq1i 2740 | . 2 ⊢ (( I ↾ 𝐴) = ∅ ↔ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅) |
4 | nel02 4345 | . . . . 5 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴) | |
5 | 4 | intnanrd 489 | . . . 4 ⊢ (𝐴 = ∅ → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)) |
6 | 5 | alrimivv 1926 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)) |
7 | ianor 983 | . . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥)) | |
8 | 7 | albii 1816 | . . . . . 6 ⊢ (∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ ∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥)) |
9 | 19.32v 1938 | . . . . . . 7 ⊢ (∀𝑦(¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥)) | |
10 | id 22 | . . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐴) | |
11 | ax6v 1966 | . . . . . . . . 9 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥 | |
12 | 11 | pm2.21i 119 | . . . . . . . 8 ⊢ (∀𝑦 ¬ 𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐴) |
13 | 10, 12 | jaoi 857 | . . . . . . 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 1808 | . . . 4 ⊢ (∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
17 | eq0 4356 | . . . 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) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∀wal 1535 = wceq 1537 ∈ wcel 2106 ∅c0 4339 {copab 5210 I cid 5582 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-res 5701 |
This theorem is referenced by: (None) |
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