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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 4356 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 | |
2 | 1 | pm2.21i 119 | . 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑) |
3 | rmo5 3396 | . 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 3374 ∃*wrmo 3375 ∅c0 4321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-dif 3950 df-nul 4322 |
This theorem is referenced by: rmosn 4722 |
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