![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 1900 | . 2 ⊢ (∃!x x = x ↔ ∃y∀x(x = x ↔ x = y)) | |
2 | equid 1586 | . . . . . 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 1356 | . . 3 ⊢ (∀x x = y ↔ ∀x(x = x ↔ x = y)) |
7 | 6 | exbii 1493 | . 2 ⊢ (∃y∀x x = y ↔ ∃y∀x(x = x ↔ x = y)) |
8 | hbae 1603 | . . 3 ⊢ (∀x x = y → ∀y∀x x = y) | |
9 | 8 | 19.9h 1531 | . 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 1240 ∃wex 1378 ∃!weu 1897 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-eu 1900 |
This theorem is referenced by: exists2 1994 |
Copyright terms: Public domain | W3C validator |