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Mirrors > Home > MPE Home > Th. List > Mathboxes > lineintmo | Structured version Visualization version GIF version |
Description: Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
lineintmo | ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an4 654 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) | |
2 | linethru 33609 | . . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → 𝐴 = (𝑥Line𝑦)) | |
3 | 2 | 3expa 1114 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝐴 = (𝑥Line𝑦)) |
4 | linethru 33609 | . . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ≠ 𝑦) → 𝐵 = (𝑥Line𝑦)) | |
5 | 4 | 3expa 1114 | . . . . . . . . . . . 12 ⊢ (((𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 ≠ 𝑦) → 𝐵 = (𝑥Line𝑦)) |
6 | eqtr3 2843 | . . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑥Line𝑦) ∧ 𝐵 = (𝑥Line𝑦)) → 𝐴 = 𝐵) | |
7 | 3, 5, 6 | syl2an 597 | . . . . . . . . . . 11 ⊢ ((((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) ∧ ((𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 ≠ 𝑦)) → 𝐴 = 𝐵) |
8 | 7 | anandirs 677 | . . . . . . . . . 10 ⊢ ((((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) ∧ 𝑥 ≠ 𝑦) → 𝐴 = 𝐵) |
9 | 8 | ex 415 | . . . . . . . . 9 ⊢ (((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥 ≠ 𝑦 → 𝐴 = 𝐵)) |
10 | 9 | necon1d 3038 | . . . . . . . 8 ⊢ (((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝐴 ≠ 𝐵 → 𝑥 = 𝑦)) |
11 | 10 | an4s 658 | . . . . . . 7 ⊢ (((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝐴 ≠ 𝐵 → 𝑥 = 𝑦)) |
12 | 1, 11 | sylan2b 595 | . . . . . 6 ⊢ (((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) → (𝐴 ≠ 𝐵 → 𝑥 = 𝑦)) |
13 | 12 | ex 415 | . . . . 5 ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐴 ≠ 𝐵 → 𝑥 = 𝑦))) |
14 | 13 | com23 86 | . . . 4 ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE) → (𝐴 ≠ 𝐵 → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = 𝑦))) |
15 | 14 | 3impia 1113 | . . 3 ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = 𝑦)) |
16 | 15 | alrimivv 1925 | . 2 ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = 𝑦)) |
17 | eleq1w 2895 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | eleq1w 2895 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
19 | 17, 18 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
20 | 19 | mo4 2646 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = 𝑦)) |
21 | 16, 20 | sylibr 236 | 1 ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∃*wmo 2616 ≠ wne 3016 (class class class)co 7150 Linecline2 33590 LinesEEclines2 33592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-ec 8285 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-ee 26671 df-btwn 26672 df-cgr 26673 df-ofs 33439 df-colinear 33495 df-ifs 33496 df-cgr3 33497 df-fs 33498 df-line2 33593 df-lines2 33595 |
This theorem is referenced by: (None) |
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