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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 44915 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
3 | n0 4310 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
4 | 2, 3 | sylibr 236 | 1 ⊢ (𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 ∀!walsc 44908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-dif 3939 df-nul 4292 df-alsc 44910 |
This theorem is referenced by: (None) |
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