![]() |
Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-euae | Structured version Visualization version GIF version |
Description: Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.) |
Ref | Expression |
---|---|
wl-euae | ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2568 | . 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤)) | |
2 | extru 1980 | . . 3 ⊢ ∃𝑥⊤ | |
3 | 2 | biantrur 532 | . 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤)) |
4 | wl-moae 36004 | . 2 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) | |
5 | 1, 3, 4 | 3bitr2i 299 | 1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∀wal 1540 ⊤wtru 1543 ∃wex 1782 ∃*wmo 2537 ∃!weu 2567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-mo 2539 df-eu 2568 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |