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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 28348 | . 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺) | |
2 | vprc 5178 | . . . . 5 ⊢ ¬ V ∈ V | |
3 | snprc 4603 | . . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅) | |
4 | 2, 3 | mpbi 233 | . . . 4 ⊢ {V} = ∅ |
5 | 4 | eqcomi 2768 | . . 3 ⊢ ∅ = {V} |
6 | 5 | eleq1i 2841 | . 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺) |
7 | 1, 6 | sylnibr 333 | 1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ∅c0 4221 {csn 4515 Pligcplig 28341 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-v 3409 df-dif 3857 df-in 3861 df-ss 3871 df-nul 4222 df-sn 4516 df-uni 4792 df-plig 28342 |
This theorem is referenced by: pliguhgr 28353 |
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