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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 28264 | . 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺) | |
2 | vprc 5183 | . . . . 5 ⊢ ¬ V ∈ V | |
3 | snprc 4613 | . . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅) | |
4 | 2, 3 | mpbi 233 | . . . 4 ⊢ {V} = ∅ |
5 | 4 | eqcomi 2807 | . . 3 ⊢ ∅ = {V} |
6 | 5 | eleq1i 2880 | . 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺) |
7 | 1, 6 | sylnibr 332 | 1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 Pligcplig 28257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-uni 4801 df-plig 28258 |
This theorem is referenced by: pliguhgr 28269 |
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