Mathbox for Wolf Lammen |
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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 2654 | . 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤)) | |
2 | extru 1980 | . . 3 ⊢ ∃𝑥⊤ | |
3 | 2 | biantrur 533 | . 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤)) |
4 | wl-moae 34758 | . 2 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) | |
5 | 1, 3, 4 | 3bitr2i 301 | 1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∀wal 1535 ⊤wtru 1538 ∃wex 1780 ∃*wmo 2620 ∃!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: (None) |
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