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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 4318 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 | |
2 | 1 | pm2.21i 119 | . 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑) |
3 | rmo5 3372 | . 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 3070 ∃!wreu 3350 ∃*wrmo 3351 ∅c0 4283 |
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-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-dif 3914 df-nul 4284 |
This theorem is referenced by: rmosn 4681 |
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