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Mirrors > Home > MPE Home > Th. List > tglndim0 | Structured version Visualization version GIF version |
Description: There are no lines in dimension 0. (Contributed by Thierry Arnoux, 18-Oct-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglndim0.d | ⊢ (𝜑 → (♯‘𝐵) = 1) |
Ref | Expression |
---|---|
tglndim0 | ⊢ (𝜑 → ¬ 𝐴 ∈ ran 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineelsb2.p | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | tglndim0.d | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) = 1) | |
3 | 2 | ad4antr 728 | . . . . 5 ⊢ (((((𝜑 ∧ 𝐴 ∈ ran 𝐿) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → (♯‘𝐵) = 1) |
4 | simpllr 772 | . . . . 5 ⊢ (((((𝜑 ∧ 𝐴 ∈ ran 𝐿) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐵) | |
5 | simplr 765 | . . . . 5 ⊢ (((((𝜑 ∧ 𝐴 ∈ ran 𝐿) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐵) | |
6 | 1, 3, 4, 5 | tgldim0eq 26216 | . . . 4 ⊢ (((((𝜑 ∧ 𝐴 ∈ ran 𝐿) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 = 𝑦) |
7 | simprr 769 | . . . 4 ⊢ (((((𝜑 ∧ 𝐴 ∈ ran 𝐿) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦) | |
8 | 6, 7 | pm2.21ddne 3098 | . . 3 ⊢ (((((𝜑 ∧ 𝐴 ∈ ran 𝐿) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → ⊥) |
9 | tglineelsb2.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
10 | tglineelsb2.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
11 | tglineelsb2.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
12 | 11 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ran 𝐿) → 𝐺 ∈ TarskiG) |
13 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ran 𝐿) → 𝐴 ∈ ran 𝐿) | |
14 | 1, 9, 10, 12, 13 | tgisline 26340 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ran 𝐿) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) |
15 | 8, 14 | r19.29vva 3333 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ran 𝐿) → ⊥) |
16 | 15 | inegd 1548 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ ran 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ⊥wfal 1540 ∈ wcel 2105 ≠ wne 3013 ran crn 5549 ‘cfv 6348 (class class class)co 7145 1c1 10526 ♯chash 13678 Basecbs 16471 TarskiGcstrkg 26143 Itvcitv 26149 LineGclng 26150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 df-trkg 26166 |
This theorem is referenced by: hpgerlem 26478 |
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