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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 35785 | . . 3 ⊢ ∃*𝑥⊥ | |
2 | nrmo.1 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑) | |
3 | 2 | imori 851 | . . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝜑) |
4 | ianor 978 | . . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝜑)) | |
5 | 3, 4 | mpbir 230 | . . . . 5 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) |
6 | 5 | bifal 1549 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ⊥) |
7 | 6 | mobii 2534 | . . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥⊥) |
8 | 1, 7 | mpbir 230 | . 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) |
9 | df-rmo 3368 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
10 | 8, 9 | mpbir 230 | 1 ⊢ ∃*𝑥 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 ⊥wfal 1545 ∈ wcel 2098 ∃*wmo 2524 ∃*wrmo 3367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-mo 2526 df-rmo 3368 |
This theorem is referenced by: (None) |
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