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Mirrors > Home > MPE Home > Th. List > rmo0 | Structured version Visualization version GIF version |
Description: Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.) |
Ref | Expression |
---|---|
rmo0 | ⊢ ∃*𝑥 ∈ ∅ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rex0 4350 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 | |
2 | 1 | pm2.21i 119 | . 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑) |
3 | rmo5 3388 | . 2 ⊢ (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)) | |
4 | 2, 3 | mpbir 230 | 1 ⊢ ∃*𝑥 ∈ ∅ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 3062 ∃!wreu 3366 ∃*wrmo 3367 ∅c0 4315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-dif 3944 df-nul 4316 |
This theorem is referenced by: rmosn 4716 |
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