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Mathbox for David A. Wheeler |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > alscn0d | Structured version Visualization version GIF version |
Description: Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.) |
Ref | Expression |
---|---|
alscn0d.1 | ⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) |
Ref | Expression |
---|---|
alscn0d | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alscn0d.1 | . . 3 ⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) | |
2 | 1 | alsc2d 48097 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
3 | n0 4341 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
4 | 2, 3 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1773 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 ∀!walsc 48090 |
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 ax-7 2003 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-ne 2935 df-dif 3946 df-nul 4318 df-alsc 48092 |
This theorem is referenced by: (None) |
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