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| Mirrors > Home > MPE Home > Th. List > Mathboxes > isline | Structured version Visualization version GIF version | ||
| Description: The predicate "is a line". (Contributed by NM, 19-Sep-2011.) | 
| Ref | Expression | 
|---|---|
| isline.l | ⊢ ≤ = (le‘𝐾) | 
| isline.j | ⊢ ∨ = (join‘𝐾) | 
| isline.a | ⊢ 𝐴 = (Atoms‘𝐾) | 
| isline.n | ⊢ 𝑁 = (Lines‘𝐾) | 
| Ref | Expression | 
|---|---|
| isline | ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isline.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 2 | isline.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
| 3 | isline.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 4 | isline.n | . . . 4 ⊢ 𝑁 = (Lines‘𝐾) | |
| 5 | 1, 2, 3, 4 | lineset 39741 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})}) | 
| 6 | 5 | eleq2d 2826 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})) | 
| 7 | 3 | fvexi 6919 | . . . . . . . 8 ⊢ 𝐴 ∈ V | 
| 8 | 7 | rabex 5338 | . . . . . . 7 ⊢ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V | 
| 9 | eleq1 2828 | . . . . . . 7 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → (𝑋 ∈ V ↔ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V)) | |
| 10 | 8, 9 | mpbiri 258 | . . . . . 6 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → 𝑋 ∈ V) | 
| 11 | 10 | adantl 481 | . . . . 5 ⊢ ((𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V) | 
| 12 | 11 | a1i 11 | . . . 4 ⊢ ((𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → ((𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V)) | 
| 13 | 12 | rexlimivv 3200 | . . 3 ⊢ (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V) | 
| 14 | eqeq1 2740 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})) | |
| 15 | 14 | anbi2d 630 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))) | 
| 16 | 15 | 2rexbidv 3221 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))) | 
| 17 | 13, 16 | elab3 3685 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})) | 
| 18 | 6, 17 | bitrdi 287 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ≠ wne 2939 ∃wrex 3069 {crab 3435 Vcvv 3479 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 lecple 17305 joincjn 18358 Atomscatm 39265 Linesclines 39497 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-lines 39504 | 
| This theorem is referenced by: islinei 39743 linepsubN 39755 isline2 39777 | 
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