Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 656 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) | |
2 | linethru 34149 | . . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → 𝐴 = (𝑥Line𝑦)) | |
3 | 2 | 3expa 1120 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝐴 = (𝑥Line𝑦)) |
4 | linethru 34149 | . . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ≠ 𝑦) → 𝐵 = (𝑥Line𝑦)) | |
5 | 4 | 3expa 1120 | . . . . . . . . . . . 12 ⊢ (((𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 ≠ 𝑦) → 𝐵 = (𝑥Line𝑦)) |
6 | eqtr3 2760 | . . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑥Line𝑦) ∧ 𝐵 = (𝑥Line𝑦)) → 𝐴 = 𝐵) | |
7 | 3, 5, 6 | syl2an 599 | . . . . . . . . . . 11 ⊢ ((((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) ∧ ((𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 ≠ 𝑦)) → 𝐴 = 𝐵) |
8 | 7 | anandirs 679 | . . . . . . . . . 10 ⊢ ((((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) ∧ 𝑥 ≠ 𝑦) → 𝐴 = 𝐵) |
9 | 8 | ex 416 | . . . . . . . . 9 ⊢ (((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥 ≠ 𝑦 → 𝐴 = 𝐵)) |
10 | 9 | necon1d 2957 | . . . . . . . 8 ⊢ (((𝐴 ∈ LinesEE ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐵 ∈ LinesEE ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝐴 ≠ 𝐵 → 𝑥 = 𝑦)) |
11 | 10 | an4s 660 | . . . . . . 7 ⊢ (((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝐴 ≠ 𝐵 → 𝑥 = 𝑦)) |
12 | 1, 11 | sylan2b 597 | . . . . . 6 ⊢ (((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) → (𝐴 ≠ 𝐵 → 𝑥 = 𝑦)) |
13 | 12 | ex 416 | . . . . 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 1119 | . . 3 ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = 𝑦)) |
16 | 15 | alrimivv 1936 | . 2 ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = 𝑦)) |
17 | eleq1w 2816 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | eleq1w 2816 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
19 | 17, 18 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
20 | 19 | mo4 2563 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = 𝑦)) |
21 | 16, 20 | sylibr 237 | 1 ⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∀wal 1541 = wceq 1543 ∈ wcel 2110 ∃*wmo 2535 ≠ wne 2935 (class class class)co 7202 Linecline2 34130 LinesEEclines2 34132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-ec 8382 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-sum 15233 df-ee 26954 df-btwn 26955 df-cgr 26956 df-ofs 33979 df-colinear 34035 df-ifs 34036 df-cgr3 34037 df-fs 34038 df-line2 34133 df-lines2 34135 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |