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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 5273. (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 2019 | . . . 4 ⊢ 𝑥 = 𝑥 | |
2 | 1 | bitru 1546 | . . 3 ⊢ (𝑥 = 𝑥 ↔ ⊤) |
3 | 2 | eubii 2670 | . 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃!𝑥⊤) |
4 | euae 2745 | . 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 1535 ⊤wtru 1538 ∃!weu 2653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-mo 2622 df-eu 2654 |
This theorem is referenced by: exists2 2747 |
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