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Mirrors > Home > MPE Home > Th. List > tgbtwnexch3 | Structured version Unicode version |
Description: Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p |
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tkgeom.d |
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tkgeom.i |
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tkgeom.g |
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tgbtwnintr.1 |
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tgbtwnintr.2 |
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tgbtwnintr.3 |
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tgbtwnintr.4 |
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tgbtwnexch3.5 |
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tgbtwnexch3.6 |
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Ref | Expression |
---|---|
tgbtwnexch3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p |
. 2
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2 | tkgeom.d |
. 2
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3 | tkgeom.i |
. 2
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4 | tkgeom.g |
. 2
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5 | tgbtwnintr.2 |
. 2
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6 | tgbtwnintr.3 |
. 2
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7 | tgbtwnintr.4 |
. 2
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8 | tgbtwnintr.1 |
. 2
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9 | tgbtwnexch3.5 |
. . 3
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10 | 1, 2, 3, 4, 8, 5, 6, 9 | tgbtwncom 23069 |
. 2
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11 | tgbtwnexch3.6 |
. . 3
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12 | 1, 2, 3, 4, 8, 6, 7, 11 | tgbtwncom 23069 |
. 2
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13 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12 | tgbtwnintr 23074 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-nul 4522 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3073 df-sbc 3288 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-br 4394 df-iota 5482 df-fv 5527 df-ov 6196 df-trkgc 23034 df-trkgb 23035 df-trkgcb 23036 df-trkg 23040 |
This theorem is referenced by: tgbtwnouttr2 23076 tgbtwnexch2 23077 tgifscgr 23090 tgcgrxfr 23099 tgbtwnconn1lem1 23134 tgbtwnconn1lem2 23135 tgbtwnconn1lem3 23136 tgbtwnconn2 23138 tgbtwnconn3 23139 tglineeltr 23169 miriso 23209 krippenlem 23220 |
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