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Mirrors > Home > MPE Home > Th. List > frcond2 | Structured version Visualization version Unicode version |
Description: The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) |
Ref | Expression |
---|---|
frcond1.v | Vtx |
frcond1.e | Edg |
Ref | Expression |
---|---|
frcond2 | FriendGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frcond1.v | . . 3 Vtx | |
2 | frcond1.e | . . 3 Edg | |
3 | 1, 2 | frcond1 27130 | . 2 FriendGraph |
4 | prex 4909 | . . . . 5 | |
5 | prex 4909 | . . . . 5 | |
6 | 4, 5 | prss 4351 | . . . 4 |
7 | 6 | bicomi 214 | . . 3 |
8 | 7 | reubii 3128 | . 2 |
9 | 3, 8 | syl6ib 241 | 1 FriendGraph |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wreu 2914 wss 3574 cpr 4179 cfv 5888 Vtxcvtx 25874 Edgcedg 25939 FriendGraph cfrgr 27120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-frgr 27121 |
This theorem is referenced by: frgreu 27132 frgrncvvdeqlem9 27171 frgr2wwlkeu 27191 numclwwlk2lem1 27235 |
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