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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nrmo | Structured version Visualization version GIF version | ||
| Description: "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.) | 
| Ref | Expression | 
|---|---|
| nrmo.1 | ⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑) | 
| Ref | Expression | 
|---|---|
| nrmo | ⊢ ∃*𝑥 ∈ 𝐴 𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mofal 36410 | . . 3 ⊢ ∃*𝑥⊥ | |
| 2 | nrmo.1 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑) | |
| 3 | 2 | imori 855 | . . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝜑) | 
| 4 | ianor 984 | . . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝜑)) | |
| 5 | 3, 4 | mpbir 231 | . . . . 5 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) | 
| 6 | 5 | bifal 1556 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ⊥) | 
| 7 | 6 | mobii 2548 | . . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥⊥) | 
| 8 | 1, 7 | mpbir 231 | . 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) | 
| 9 | df-rmo 3380 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 10 | 8, 9 | mpbir 231 | 1 ⊢ ∃*𝑥 ∈ 𝐴 𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ⊥wfal 1552 ∈ wcel 2108 ∃*wmo 2538 ∃*wrmo 3379 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-mo 2540 df-rmo 3380 | 
| This theorem is referenced by: (None) | 
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