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Mirrors > Home > MPE Home > Th. List > tgbtwnconn22 | Structured version Visualization version GIF version |
Description: Double connectivity law for betweenness. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
Ref | Expression |
---|---|
tgbtwnconn.p | ⊢ 𝑃 = (Base‘𝐺) |
tgbtwnconn.i | ⊢ 𝐼 = (Itv‘𝐺) |
tgbtwnconn.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwnconn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwnconn.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwnconn.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgbtwnconn.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgbtwnconn22.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
tgbtwnconn22.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
tgbtwnconn22.2 | ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
tgbtwnconn22.3 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
tgbtwnconn22.4 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
tgbtwnconn22.5 | ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐸)) |
Ref | Expression |
---|---|
tgbtwnconn22 | ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgbtwnconn.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2610 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | tgbtwnconn.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tgbtwnconn.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐺 ∈ TarskiG) |
6 | tgbtwnconn.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷 ∈ 𝑃) |
8 | tgbtwnconn.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ 𝑃) |
10 | tgbtwnconn.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ 𝑃) |
12 | tgbtwnconn22.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐸 ∈ 𝑃) |
14 | tgbtwnconn22.2 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵) | |
15 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ≠ 𝐵) |
16 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷)) | |
17 | 1, 2, 3, 5, 11, 9, 7, 16 | tgbtwncom 25183 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐷𝐼𝐵)) |
18 | tgbtwnconn22.5 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐸)) | |
19 | 18 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ (𝐶𝐼𝐸)) |
20 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19 | tgbtwnouttr2 25190 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ (𝐷𝐼𝐸)) |
21 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG) |
22 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ 𝑃) |
23 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ 𝑃) |
24 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐸 ∈ 𝑃) |
25 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶 ∈ 𝑃) |
26 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶)) | |
27 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐸)) |
28 | 1, 2, 3, 21, 25, 23, 24, 27 | tgbtwncom 25183 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ (𝐸𝐼𝐶)) |
29 | 1, 2, 3, 21, 22, 23, 24, 25, 26, 28 | tgbtwnintr 25188 | . 2 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ (𝐷𝐼𝐸)) |
30 | tgbtwnconn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
31 | tgbtwnconn22.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
32 | tgbtwnconn22.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
33 | tgbtwnconn22.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
34 | 1, 3, 4, 30, 10, 8, 6, 31, 32, 33 | tgbtwnconn2 25271 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) |
35 | 20, 29, 34 | mpjaodan 823 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 |
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-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 df-cgrg 25206 |
This theorem is referenced by: mideulem2 25426 |
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