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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 5270. (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 2015 | . . . 4 ⊢ 𝑥 = 𝑥 | |
2 | 1 | bitru 1542 | . . 3 ⊢ (𝑥 = 𝑥 ↔ ⊤) |
3 | 2 | eubii 2666 | . 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃!𝑥⊤) |
4 | euae 2743 | . 2 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) | |
5 | 3, 4 | bitri 277 | 1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1531 ⊤wtru 1534 ∃!weu 2649 |
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 1907 ax-6 1966 ax-7 2011 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1536 df-ex 1777 df-mo 2618 df-eu 2650 |
This theorem is referenced by: exists2 2745 |
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