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Mirrors > Home > MPE Home > Th. List > usgrprc | Structured version Visualization version GIF version |
Description: The class of simple graphs is a proper class (and therefore, because of prcssprc 5318, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
usgrprc | β’ USGraph β V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 β’ {β¨π£, πβ© β£ π:β βΆβ } = {β¨π£, πβ© β£ π:β βΆβ } | |
2 | 1 | griedg0ssusgr 29026 | . 2 β’ {β¨π£, πβ© β£ π:β βΆβ } β USGraph |
3 | 1 | griedg0prc 29025 | . 2 β’ {β¨π£, πβ© β£ π:β βΆβ } β V |
4 | prcssprc 5318 | . 2 β’ (({β¨π£, πβ© β£ π:β βΆβ } β USGraph β§ {β¨π£, πβ© β£ π:β βΆβ } β V) β USGraph β V) | |
5 | 2, 3, 4 | mp2an 689 | 1 β’ USGraph β V |
Colors of variables: wff setvar class |
Syntax hints: β wnel 3040 Vcvv 3468 β wss 3943 β c0 4317 {copab 5203 βΆwf 6532 USGraphcusgr 28913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fv 6544 df-2nd 7972 df-iedg 28763 df-usgr 28915 |
This theorem is referenced by: (None) |
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