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Mirrors > Home > MPE Home > Th. List > l2p | Structured version Visualization version GIF version |
Description: For any line in a planar incidence geometry, there exist two different points on the line. (Contributed by AV, 28-Nov-2021.) |
Ref | Expression |
---|---|
l2p.1 | ⊢ 𝑃 = ∪ 𝐺 |
Ref | Expression |
---|---|
l2p | ⊢ ((𝐺 ∈ Plig ∧ 𝐿 ∈ 𝐺) → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | l2p.1 | . . . . 5 ⊢ 𝑃 = ∪ 𝐺 | |
2 | 1 | isplig 28045 | . . . 4 ⊢ (𝐺 ∈ Plig → (𝐺 ∈ Plig ↔ (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝐺 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝐺 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∀𝑙 ∈ 𝐺 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙)))) |
3 | eleq2 2856 | . . . . . . . 8 ⊢ (𝑙 = 𝐿 → (𝑎 ∈ 𝑙 ↔ 𝑎 ∈ 𝐿)) | |
4 | eleq2 2856 | . . . . . . . 8 ⊢ (𝑙 = 𝐿 → (𝑏 ∈ 𝑙 ↔ 𝑏 ∈ 𝐿)) | |
5 | 3, 4 | 3anbi23d 1419 | . . . . . . 7 ⊢ (𝑙 = 𝐿 → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ↔ (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿))) |
6 | 5 | 2rexbidv 3247 | . . . . . 6 ⊢ (𝑙 = 𝐿 → (∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ↔ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿))) |
7 | 6 | rspccv 3534 | . . . . 5 ⊢ (∀𝑙 ∈ 𝐺 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) → (𝐿 ∈ 𝐺 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿))) |
8 | 7 | 3ad2ant2 1115 | . . . 4 ⊢ ((∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝐺 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝐺 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∀𝑙 ∈ 𝐺 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙)) → (𝐿 ∈ 𝐺 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿))) |
9 | 2, 8 | syl6bi 245 | . . 3 ⊢ (𝐺 ∈ Plig → (𝐺 ∈ Plig → (𝐿 ∈ 𝐺 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿)))) |
10 | 9 | pm2.43i 52 | . 2 ⊢ (𝐺 ∈ Plig → (𝐿 ∈ 𝐺 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿))) |
11 | 10 | imp 398 | 1 ⊢ ((𝐺 ∈ Plig ∧ 𝐿 ∈ 𝐺) → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2969 ∀wral 3090 ∃wrex 3091 ∃!wreu 3092 ∪ cuni 4717 Pligcplig 28043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2752 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ral 3095 df-rex 3096 df-reu 3097 df-v 3419 df-uni 4718 df-plig 28044 |
This theorem is referenced by: nsnlplig 28050 nsnlpligALT 28051 n0lpligALT 28053 |
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