![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4271 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 | |
2 | 1 | pm2.21i 119 | . 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑) |
3 | rmo5 3379 | . 2 ⊢ (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)) | |
4 | 2, 3 | mpbir 234 | 1 ⊢ ∃*𝑥 ∈ ∅ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 3107 ∃!wreu 3108 ∃*wrmo 3109 ∅c0 4243 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-dif 3884 df-nul 4244 |
This theorem is referenced by: rmosn 4615 |
Copyright terms: Public domain | W3C validator |