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Mirrors > Home > MPE Home > Th. List > uhgr0 | Structured version Visualization version Unicode version |
Description: The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.) |
Ref | Expression |
---|---|
uhgr0 | UHGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0 6086 | . . 3 | |
2 | dm0 5339 | . . . 4 | |
3 | pw0 4343 | . . . . . 6 | |
4 | 3 | difeq1i 3724 | . . . . 5 |
5 | difid 3948 | . . . . 5 | |
6 | 4, 5 | eqtri 2644 | . . . 4 |
7 | 2, 6 | feq23i 6039 | . . 3 |
8 | 1, 7 | mpbir 221 | . 2 |
9 | 0ex 4790 | . . 3 | |
10 | vtxval0 25931 | . . . . 5 Vtx | |
11 | 10 | eqcomi 2631 | . . . 4 Vtx |
12 | iedgval0 25932 | . . . . 5 iEdg | |
13 | 12 | eqcomi 2631 | . . . 4 iEdg |
14 | 11, 13 | isuhgr 25955 | . . 3 UHGraph |
15 | 9, 14 | ax-mp 5 | . 2 UHGraph |
16 | 8, 15 | mpbir 221 | 1 UHGraph |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wcel 1990 cvv 3200 cdif 3571 c0 3915 cpw 4158 csn 4177 cdm 5114 wf 5884 cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 UHGraph cuhgr 25951 |
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-8 1992 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-pow 4843 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-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-slot 15861 df-base 15863 df-edgf 25868 df-vtx 25876 df-iedg 25877 df-uhgr 25953 |
This theorem is referenced by: (None) |
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