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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 5558 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)) | |
| 2 | opabresid 6067 | . . 3 ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
| 3 | 2 | eqeq1i 2741 | . 2 ⊢ (( I ↾ 𝐴) = ∅ ↔ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ∅) | 
| 4 | nel02 4338 | . . . . 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 983 | . . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 = 𝑥)) | |
| 8 | 7 | albii 1818 | . . . . . 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 1967 | . . . . . . . . 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 1810 | . . . 4 ⊢ (∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥 ∈ 𝐴) | 
| 17 | eq0 4349 | . . . 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 1537 = wceq 1539 ∈ wcel 2107 ∅c0 4332 {copab 5204 I cid 5576 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-res 5696 | 
| This theorem is referenced by: (None) | 
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