![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > n0lplig | Structured version Visualization version GIF version |
Description: There is no "empty line" in a planar incidence geometry. (Contributed by AV, 28-Nov-2021.) (Proof shortened by BJ, 2-Dec-2021.) |
Ref | Expression |
---|---|
n0lplig | ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsnlplig 29465 | . 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺) | |
2 | vprc 5273 | . . . . 5 ⊢ ¬ V ∈ V | |
3 | snprc 4679 | . . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅) | |
4 | 2, 3 | mpbi 229 | . . . 4 ⊢ {V} = ∅ |
5 | 4 | eqcomi 2742 | . . 3 ⊢ ∅ = {V} |
6 | 5 | eleq1i 2825 | . 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺) |
7 | 1, 6 | sylnibr 329 | 1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 {csn 4587 Pligcplig 29458 |
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-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-v 3446 df-dif 3914 df-in 3918 df-ss 3928 df-nul 4284 df-sn 4588 df-uni 4867 df-plig 29459 |
This theorem is referenced by: pliguhgr 29470 |
Copyright terms: Public domain | W3C validator |