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Mirrors > Home > MPE Home > Th. List > reu0 | Structured version Visualization version GIF version |
Description: Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.) |
Ref | Expression |
---|---|
reu0 | ⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rex0 4355 | . 2 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 | |
2 | reurex 3381 | . 2 ⊢ (∃!𝑥 ∈ ∅ 𝜑 → ∃𝑥 ∈ ∅ 𝜑) | |
3 | 1, 2 | mto 196 | 1 ⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∃wrex 3071 ∃!wreu 3375 ∅c0 4320 |
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 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-dif 3949 df-nul 4321 |
This theorem is referenced by: join0 18345 meet0 18346 |
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