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Mirrors > Home > MPE Home > Th. List > tglineineq | Structured version Visualization version GIF version |
Description: Two distinct lines intersect in at most one point, variation. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 6-Aug-2019.) |
Ref | Expression |
---|---|
tglineintmo.p | ⊢ 𝑃 = (Base‘𝐺) |
tglineintmo.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineintmo.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineintmo.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglineintmo.a | ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) |
tglineintmo.b | ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) |
tglineintmo.c | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
tglineineq.x | ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
tglineineq.y | ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∩ 𝐵)) |
Ref | Expression |
---|---|
tglineineq | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineineq.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) | |
2 | tglineineq.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∩ 𝐵)) | |
3 | tglineintmo.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
4 | tglineintmo.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tglineintmo.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | tglineintmo.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | tglineintmo.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | |
8 | tglineintmo.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) | |
9 | tglineintmo.c | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
10 | 3, 4, 5, 6, 7, 8, 9 | tglineintmo 28566 | . 2 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
11 | elin 3962 | . . 3 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) | |
12 | 1, 11 | sylib 217 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) |
13 | elin 3962 | . . 3 ⊢ (𝑌 ∈ (𝐴 ∩ 𝐵) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) | |
14 | 2, 13 | sylib 217 | . 2 ⊢ (𝜑 → (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) |
15 | eleq1 2814 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) | |
16 | eleq1 2814 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) | |
17 | 15, 16 | anbi12d 630 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵))) |
18 | eleq1 2814 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴)) | |
19 | eleq1 2814 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ 𝐵 ↔ 𝑌 ∈ 𝐵)) | |
20 | 18, 19 | anbi12d 630 | . . 3 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))) |
21 | 17, 20 | moi 3711 | . 2 ⊢ (((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ 𝑌 ∈ (𝐴 ∩ 𝐵)) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))) → 𝑋 = 𝑌) |
22 | 1, 2, 10, 12, 14, 21 | syl212anc 1377 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃*wmo 2527 ≠ wne 2930 ∩ cin 3945 ran crn 5675 ‘cfv 6546 Basecbs 17208 TarskiGcstrkg 28351 Itvcitv 28357 LineGclng 28358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-oadd 8492 df-er 8726 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-dju 9937 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-xnn0 12591 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-hash 14343 df-word 14518 df-concat 14574 df-s1 14599 df-s2 14852 df-s3 14853 df-trkgc 28372 df-trkgb 28373 df-trkgcb 28374 df-trkg 28377 df-cgrg 28435 |
This theorem is referenced by: isperp2 28639 footne 28647 lnopp2hpgb 28687 colopp 28693 lmieu 28708 |
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