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Mirrors > Home > MPE Home > Th. List > nsnlplig | Structured version Visualization version GIF version |
Description: There is no "one-point line" in a planar incidence geometry. (Contributed by BJ, 2-Dec-2021.) (Proof shortened by AV, 5-Dec-2021.) |
Ref | Expression |
---|---|
nsnlplig | ⊢ (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺 | |
2 | 1 | l2p 28262 | . . 3 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) |
3 | elsni 4542 | . . . . . . . 8 ⊢ (𝑎 ∈ {𝐴} → 𝑎 = 𝐴) | |
4 | elsni 4542 | . . . . . . . 8 ⊢ (𝑏 ∈ {𝐴} → 𝑏 = 𝐴) | |
5 | eqtr3 2820 | . . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏) | |
6 | eqneqall 2998 | . . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → ¬ {𝐴} ∈ 𝐺)) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → ¬ {𝐴} ∈ 𝐺)) |
8 | 3, 4, 7 | syl2an 598 | . . . . . . 7 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → (𝑎 ≠ 𝑏 → ¬ {𝐴} ∈ 𝐺)) |
9 | 8 | impcom 411 | . . . . . 6 ⊢ ((𝑎 ≠ 𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → ¬ {𝐴} ∈ 𝐺) |
10 | 9 | 3impb 1112 | . . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺) |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)) |
12 | 11 | rexlimivv 3251 | . . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺) |
13 | 2, 12 | syl 17 | . 2 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ¬ {𝐴} ∈ 𝐺) |
14 | 13 | pm2.01da 798 | 1 ⊢ (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {csn 4525 ∪ cuni 4800 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 |
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-in 3888 df-ss 3898 df-sn 4526 df-uni 4801 df-plig 28258 |
This theorem is referenced by: n0lplig 28266 |
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