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Mirrors > Home > MPE Home > Th. List > nrexrmo | Structured version Visualization version GIF version |
Description: Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.) |
Ref | Expression |
---|---|
nrexrmo | ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.21 123 | . 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
2 | rmo5 3432 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
3 | 1, 2 | sylibr 235 | 1 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∃wrex 3136 ∃!wreu 3137 ∃*wrmo 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-mo 2615 df-eu 2647 df-rex 3141 df-reu 3142 df-rmo 3143 |
This theorem is referenced by: (None) |
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