|   | 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 4359 | . . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 | |
| 2 | 1 | pm2.21i 119 | . 2 ⊢ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑) | 
| 3 | rmo5 3399 | . 2 ⊢ (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)) | |
| 4 | 2, 3 | mpbir 231 | 1 ⊢ ∃*𝑥 ∈ ∅ 𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∃wrex 3069 ∃!wreu 3377 ∃*wrmo 3378 ∅c0 4332 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-dif 3953 df-nul 4333 | 
| This theorem is referenced by: rmosn 4718 | 
| Copyright terms: Public domain | W3C validator |