![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > griedg0ssusgr | Structured version Visualization version GIF version |
Description: The class of all simple graphs is a superclass of the class of empty graphs represented as ordered pairs. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
griedg0prc.u | β’ π = {β¨π£, πβ© β£ π:β βΆβ } |
Ref | Expression |
---|---|
griedg0ssusgr | β’ π β USGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | griedg0prc.u | . . . . 5 β’ π = {β¨π£, πβ© β£ π:β βΆβ } | |
2 | 1 | eleq2i 2826 | . . . 4 β’ (π β π β π β {β¨π£, πβ© β£ π:β βΆβ }) |
3 | elopab 5485 | . . . 4 β’ (π β {β¨π£, πβ© β£ π:β βΆβ } β βπ£βπ(π = β¨π£, πβ© β§ π:β βΆβ )) | |
4 | 2, 3 | bitri 275 | . . 3 β’ (π β π β βπ£βπ(π = β¨π£, πβ© β§ π:β βΆβ )) |
5 | opex 5422 | . . . . . . . 8 β’ β¨π£, πβ© β V | |
6 | 5 | a1i 11 | . . . . . . 7 β’ (π:β βΆβ β β¨π£, πβ© β V) |
7 | vex 3448 | . . . . . . . . 9 β’ π£ β V | |
8 | vex 3448 | . . . . . . . . 9 β’ π β V | |
9 | 7, 8 | opiedgfvi 28003 | . . . . . . . 8 β’ (iEdgββ¨π£, πβ©) = π |
10 | f0bi 6726 | . . . . . . . . 9 β’ (π:β βΆβ β π = β ) | |
11 | 10 | biimpi 215 | . . . . . . . 8 β’ (π:β βΆβ β π = β ) |
12 | 9, 11 | eqtrid 2785 | . . . . . . 7 β’ (π:β βΆβ β (iEdgββ¨π£, πβ©) = β ) |
13 | 6, 12 | usgr0e 28226 | . . . . . 6 β’ (π:β βΆβ β β¨π£, πβ© β USGraph) |
14 | 13 | adantl 483 | . . . . 5 β’ ((π = β¨π£, πβ© β§ π:β βΆβ ) β β¨π£, πβ© β USGraph) |
15 | eleq1 2822 | . . . . . 6 β’ (π = β¨π£, πβ© β (π β USGraph β β¨π£, πβ© β USGraph)) | |
16 | 15 | adantr 482 | . . . . 5 β’ ((π = β¨π£, πβ© β§ π:β βΆβ ) β (π β USGraph β β¨π£, πβ© β USGraph)) |
17 | 14, 16 | mpbird 257 | . . . 4 β’ ((π = β¨π£, πβ© β§ π:β βΆβ ) β π β USGraph) |
18 | 17 | exlimivv 1936 | . . 3 β’ (βπ£βπ(π = β¨π£, πβ© β§ π:β βΆβ ) β π β USGraph) |
19 | 4, 18 | sylbi 216 | . 2 β’ (π β π β π β USGraph) |
20 | 19 | ssriv 3949 | 1 β’ π β USGraph |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 Vcvv 3444 β wss 3911 β c0 4283 β¨cop 4593 {copab 5168 βΆwf 6493 βcfv 6497 iEdgciedg 27990 USGraphcusgr 28142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fv 6505 df-2nd 7923 df-iedg 27992 df-usgr 28144 |
This theorem is referenced by: usgrprc 28256 |
Copyright terms: Public domain | W3C validator |