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Mirrors > Home > MPE Home > Th. List > cusgrauvtxb | Structured version Visualization version Unicode version |
Description: An undirected simple graph is complete if and only if each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) |
Ref | Expression |
---|---|
cusgrauvtxb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrav 25113 |
. 2
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2 | iscusgra 25232 |
. . . 4
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3 | 2 | baibd 925 |
. . 3
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4 | dfcleq 2455 |
. . . . 5
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5 | 4 | a1i 11 |
. . . 4
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6 | isuvtx 25264 |
. . . . . 6
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7 | 6 | eqeq1d 2463 |
. . . . 5
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8 | 7 | adantr 471 |
. . . 4
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9 | df-ral 2753 |
. . . . 5
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10 | bicom 205 |
. . . . . . . 8
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11 | 10 | a1i 11 |
. . . . . . 7
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12 | pm4.71 640 |
. . . . . . . 8
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13 | 12 | a1i 11 |
. . . . . . 7
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14 | sneq 3989 |
. . . . . . . . . . . 12
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15 | 14 | difeq2d 3562 |
. . . . . . . . . . 11
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16 | preq2 4064 |
. . . . . . . . . . . 12
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17 | 16 | eleq1d 2523 |
. . . . . . . . . . 11
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18 | 15, 17 | raleqbidv 3012 |
. . . . . . . . . 10
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19 | 18 | elrab 3207 |
. . . . . . . . 9
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20 | 19 | a1i 11 |
. . . . . . . 8
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21 | 20 | bibi1d 325 |
. . . . . . 7
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22 | 11, 13, 21 | 3bitr4d 293 |
. . . . . 6
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23 | 22 | albidv 1777 |
. . . . 5
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24 | 9, 23 | syl5bb 265 |
. . . 4
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25 | 5, 8, 24 | 3bitr4rd 294 |
. . 3
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26 | 3, 25 | bitrd 261 |
. 2
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27 | 1, 26 | mpancom 680 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pr 4652 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-sbc 3279 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-nul 3743 df-if 3893 df-sn 3980 df-pr 3982 df-op 3986 df-uni 4212 df-br 4416 df-opab 4475 df-id 4767 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-iota 5564 df-fun 5602 df-fv 5608 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-usgra 25108 df-cusgra 25197 df-uvtx 25198 |
This theorem is referenced by: vdiscusgra 25697 |
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