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Mathbox for Norm Megill |
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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 37034 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})}) |
6 | 5 | eleq2d 2875 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})) |
7 | 3 | fvexi 6659 | . . . . . . . 8 ⊢ 𝐴 ∈ V |
8 | 7 | rabex 5199 | . . . . . . 7 ⊢ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V |
9 | eleq1 2877 | . . . . . . 7 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → (𝑋 ∈ V ↔ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V)) | |
10 | 8, 9 | mpbiri 261 | . . . . . 6 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → 𝑋 ∈ V) |
11 | 10 | adantl 485 | . . . . 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 3251 | . . 3 ⊢ (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V) |
14 | eqeq1 2802 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})) | |
15 | 14 | anbi2d 631 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))) |
16 | 15 | 2rexbidv 3259 | . . 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 3622 | . 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 | syl6bb 290 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ≠ wne 2987 ∃wrex 3107 {crab 3110 Vcvv 3441 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 lecple 16564 joincjn 17546 Atomscatm 36559 Linesclines 36790 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
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-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-lines 36797 |
This theorem is referenced by: islinei 37036 linepsubN 37048 isline2 37070 |
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