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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 3376 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∃wrex 3074 ∃!wreu 3354 ∃*wrmo 3355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-mo 2539 df-eu 2568 df-rex 3075 df-rmo 3356 df-reu 3357 |
This theorem is referenced by: (None) |
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