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Mirrors > Home > MPE Home > Th. List > 4cyclusnfrgra | Structured version Visualization version Unicode version |
Description: A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) |
Ref | Expression |
---|---|
4cyclusnfrgra |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 772 |
. . . . 5
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2 | simprr 774 |
. . . . 5
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3 | simpl3 1035 |
. . . . 5
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4 | 4cycl2vnunb 25824 |
. . . . 5
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5 | 1, 2, 3, 4 | syl3anc 1292 |
. . . 4
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6 | frgraunss 25802 |
. . . . . . . 8
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7 | pm2.24 112 |
. . . . . . . 8
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8 | 6, 7 | syl6com 35 |
. . . . . . 7
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9 | 8 | 3ad2ant2 1052 |
. . . . . 6
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10 | 9 | com23 80 |
. . . . 5
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11 | 10 | adantr 472 |
. . . 4
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12 | 5, 11 | mpd 15 |
. . 3
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13 | 12 | pm2.01d 174 |
. 2
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14 | 13 | ex 441 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-br 4396 df-opab 4455 df-xp 4845 df-rel 4846 df-cnv 4847 df-dm 4849 df-rn 4850 df-frgra 25796 |
This theorem is referenced by: frgranbnb 25827 frgrawopreg 25856 |
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