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Mirrors > Home > MPE Home > Th. List > usgraedgrnv | Structured version Unicode version |
Description: An edge of an undirected simple graph always connects two vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.) |
Ref | Expression |
---|---|
usgraedgrnv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgraf0 23427 |
. . 3
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2 | f1f 5713 |
. . 3
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3 | df-f 5529 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | ssel2 3458 |
. . . . . . 7
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5 | fveq2 5798 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | eqeq1d 2456 |
. . . . . . . . 9
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7 | 6 | elrab 3222 |
. . . . . . . 8
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8 | prex 4641 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | elpw 3973 |
. . . . . . . . . 10
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10 | ianor 488 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | elprchashprn2 12273 |
. . . . . . . . . . . . . . 15
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | elprchashprn2 12273 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | prcom 4060 |
. . . . . . . . . . . . . . . . . . 19
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | fveq2d 5802 |
. . . . . . . . . . . . . . . . 17
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16 | 15 | eqeq1d 2456 |
. . . . . . . . . . . . . . . 16
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17 | 12, 16 | mtbid 300 |
. . . . . . . . . . . . . . 15
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18 | 11, 17 | jaoi 379 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 10, 18 | sylbi 195 |
. . . . . . . . . . . . 13
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20 | 19 | con4i 130 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | prssg 4135 |
. . . . . . . . . . . . 13
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22 | 21 | biimprd 223 |
. . . . . . . . . . . 12
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23 | 20, 22 | syl 16 |
. . . . . . . . . . 11
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24 | 23 | com12 31 |
. . . . . . . . . 10
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25 | 9, 24 | sylbi 195 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | imp 429 |
. . . . . . . 8
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27 | 7, 26 | sylbi 195 |
. . . . . . 7
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28 | 4, 27 | syl 16 |
. . . . . 6
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29 | 28 | ex 434 |
. . . . 5
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30 | 29 | adantl 466 |
. . . 4
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31 | 3, 30 | sylbi 195 |
. . 3
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32 | 1, 2, 31 | 3syl 20 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 32 | imp 429 |
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-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 ax-cnex 9448 ax-resscn 9449 ax-1cn 9450 ax-icn 9451 ax-addcl 9452 ax-addrcl 9453 ax-mulcl 9454 ax-mulrcl 9455 ax-mulcom 9456 ax-addass 9457 ax-mulass 9458 ax-distr 9459 ax-i2m1 9460 ax-1ne0 9461 ax-1rid 9462 ax-rnegex 9463 ax-rrecex 9464 ax-cnre 9465 ax-pre-lttri 9466 ax-pre-lttrn 9467 ax-pre-ltadd 9468 ax-pre-mulgt0 9469 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-nel 2650 df-ral 2803 df-rex 2804 df-reu 2805 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-pss 3451 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-tp 3989 df-op 3991 df-uni 4199 df-int 4236 df-iun 4280 df-br 4400 df-opab 4458 df-mpt 4459 df-tr 4493 df-eprel 4739 df-id 4743 df-po 4748 df-so 4749 df-fr 4786 df-we 4788 df-ord 4829 df-on 4830 df-lim 4831 df-suc 4832 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-riota 6160 df-ov 6202 df-oprab 6203 df-mpt2 6204 df-om 6586 df-1st 6686 df-2nd 6687 df-recs 6941 df-rdg 6975 df-1o 7029 df-er 7210 df-en 7420 df-dom 7421 df-sdom 7422 df-fin 7423 df-card 8219 df-pnf 9530 df-mnf 9531 df-xr 9532 df-ltxr 9533 df-le 9534 df-sub 9707 df-neg 9708 df-nn 10433 df-2 10490 df-n0 10690 df-z 10757 df-uz 10972 df-fz 11554 df-hash 12220 df-usgra 23417 |
This theorem is referenced by: nbusgra 23490 nbgra0nb 23491 nbgraeledg 23492 nbgraisvtx 23493 constr3trllem2 23688 constr3trllem5 23691 usgra2adedgspthlem2 30451 usgra2adedgspth 30452 usgra2adedgwlk 30453 usgra2adedgwlkon 30454 frgranbnb 30759 frgraeu 30794 extwwlkfablem1 30814 numclwwlkun 30819 |
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