![]() |
Mathbox for David A. Wheeler |
< Previous
Next >
Nearby theorems |
|
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 47327 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
3 | n0 4307 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
4 | 2, 3 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1782 ∈ wcel 2107 ≠ wne 2940 ∅c0 4283 ∀!walsc 47320 |
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 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-ne 2941 df-dif 3914 df-nul 4284 df-alsc 47322 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |