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Mirrors > Home > MPE Home > Th. List > exists1 | Structured version Visualization version GIF version |
Description: Two ways to express "exactly one thing exists". The left-hand side requires only one variable to express this. Both sides are false in set theory, see Theorem dtru 5447. (Contributed by NM, 5-Apr-2004.) (Proof shortened by BJ, 7-Oct-2022.) |
Ref | Expression |
---|---|
exists1 | ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2009 | . . . 4 ⊢ 𝑥 = 𝑥 | |
2 | 1 | bitru 1546 | . . 3 ⊢ (𝑥 = 𝑥 ↔ ⊤) |
3 | 2 | eubii 2583 | . 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃!𝑥⊤) |
4 | euae 2658 | . 2 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) | |
5 | 3, 4 | bitri 275 | 1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wal 1535 ⊤wtru 1538 ∃!weu 2566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-mo 2538 df-eu 2567 |
This theorem is referenced by: exists2 2660 |
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