![]() |
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 30311 | . 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺) | |
2 | vprc 5319 | . . . . 5 ⊢ ¬ V ∈ V | |
3 | snprc 4726 | . . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅) | |
4 | 2, 3 | mpbi 229 | . . . 4 ⊢ {V} = ∅ |
5 | 4 | eqcomi 2737 | . . 3 ⊢ ∅ = {V} |
6 | 5 | eleq1i 2820 | . 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺) |
7 | 1, 6 | sylnibr 328 | 1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∅c0 4326 {csn 4632 Pligcplig 30304 |
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-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-v 3475 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4327 df-sn 4633 df-uni 4913 df-plig 30305 |
This theorem is referenced by: pliguhgr 30316 |
Copyright terms: Public domain | W3C validator |