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 46202 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
3 | n0 4277 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
4 | 2, 3 | sylibr 237 | 1 ⊢ (𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1787 ∈ wcel 2112 ≠ wne 2942 ∅c0 4253 ∀!walsc 46195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-ne 2943 df-dif 3885 df-nul 4254 df-alsc 46197 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |