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Mirrors > Home > ILE Home > Th. List > exists1 | GIF version |
Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory. (Contributed by NM, 5-Apr-2004.) |
Ref | Expression |
---|---|
exists1 | ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 1903 | . 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦)) | |
2 | equid 1589 | . . . . . 6 ⊢ 𝑥 = 𝑥 | |
3 | 2 | tbt 236 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑦 ↔ 𝑥 = 𝑥)) |
4 | bicom 128 | . . . . 5 ⊢ ((𝑥 = 𝑦 ↔ 𝑥 = 𝑥) ↔ (𝑥 = 𝑥 ↔ 𝑥 = 𝑦)) | |
5 | 3, 4 | bitri 173 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑥 ↔ 𝑥 = 𝑦)) |
6 | 5 | albii 1359 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦)) |
7 | 6 | exbii 1496 | . 2 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∃𝑦∀𝑥(𝑥 = 𝑥 ↔ 𝑥 = 𝑦)) |
8 | hbae 1606 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | |
9 | 8 | 19.9h 1534 | . 2 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦) |
10 | 1, 7, 9 | 3bitr2i 197 | 1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wal 1241 ∃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-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-eu 1903 |
This theorem is referenced by: exists2 1997 |
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