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Mirrors > Home > ILE Home > Th. List > exists2 | GIF version |
Description: A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
exists2 | ⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbeu1 1910 | . . . . . 6 ⊢ (∃!𝑥 𝑥 = 𝑥 → ∀𝑥∃!𝑥 𝑥 = 𝑥) | |
2 | hba1 1433 | . . . . . 6 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
3 | exists1 1996 | . . . . . . 7 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | |
4 | ax16 1694 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | |
5 | 3, 4 | sylbi 114 | . . . . . 6 ⊢ (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑)) |
6 | 1, 2, 5 | exlimdh 1487 | . . . . 5 ⊢ (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
7 | 6 | com12 27 | . . . 4 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑)) |
8 | alexim 1536 | . . . 4 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑) | |
9 | 7, 8 | syl6 29 | . . 3 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑)) |
10 | 9 | con2d 554 | . 2 ⊢ (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥)) |
11 | 10 | imp 115 | 1 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∀wal 1241 = wceq 1243 ∃wex 1381 ∃!weu 1900 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 |
This theorem is referenced by: (None) |
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