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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 34042 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})}) |
6 | 5 | eleq2d 2673 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})) |
7 | fvex 6113 | . . . . . . . . 9 ⊢ (Atoms‘𝐾) ∈ V | |
8 | 3, 7 | eqeltri 2684 | . . . . . . . 8 ⊢ 𝐴 ∈ V |
9 | 8 | rabex 4740 | . . . . . . 7 ⊢ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V |
10 | eleq1 2676 | . . . . . . 7 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → (𝑋 ∈ V ↔ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V)) | |
11 | 9, 10 | mpbiri 247 | . . . . . 6 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → 𝑋 ∈ V) |
12 | 11 | adantl 481 | . . . . 5 ⊢ ((𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V) |
13 | 12 | a1i 11 | . . . 4 ⊢ ((𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → ((𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V)) |
14 | 13 | rexlimivv 3018 | . . 3 ⊢ (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V) |
15 | eqeq1 2614 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})) | |
16 | 15 | anbi2d 736 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))) |
17 | 16 | 2rexbidv 3039 | . . 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)}))) |
18 | 14, 17 | elab3 3327 | . 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)})) |
19 | 6, 18 | syl6bb 275 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∃wrex 2897 {crab 2900 Vcvv 3173 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 lecple 15775 joincjn 16767 Atomscatm 33568 Linesclines 33798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-lines 33805 |
This theorem is referenced by: islinei 34044 linepsubN 34056 isline2 34078 |
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