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Mirrors > Home > MPE Home > Th. List > griedg0prc | Structured version Visualization version GIF version |
Description: The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
griedg0prc.u | β’ π = {β¨π£, πβ© β£ π:β βΆβ } |
Ref | Expression |
---|---|
griedg0prc | β’ π β V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5300 | . . . 4 β’ β β V | |
2 | feq1 6692 | . . . 4 β’ (π = β β (π:β βΆβ β β :β βΆβ )) | |
3 | f0 6766 | . . . 4 β’ β :β βΆβ | |
4 | 1, 2, 3 | ceqsexv2d 3523 | . . 3 β’ βπ π:β βΆβ |
5 | opabn1stprc 8043 | . . 3 β’ (βπ π:β βΆβ β {β¨π£, πβ© β£ π:β βΆβ } β V) | |
6 | 4, 5 | ax-mp 5 | . 2 β’ {β¨π£, πβ© β£ π:β βΆβ } β V |
7 | griedg0prc.u | . . 3 β’ π = {β¨π£, πβ© β£ π:β βΆβ } | |
8 | neleq1 3046 | . . 3 β’ (π = {β¨π£, πβ© β£ π:β βΆβ } β (π β V β {β¨π£, πβ© β£ π:β βΆβ } β V)) | |
9 | 7, 8 | ax-mp 5 | . 2 β’ (π β V β {β¨π£, πβ© β£ π:β βΆβ } β V) |
10 | 6, 9 | mpbir 230 | 1 β’ π β V |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1533 βwex 1773 β wnel 3040 Vcvv 3468 β c0 4317 {copab 5203 βΆwf 6533 |
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 7722 |
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-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-fun 6539 df-fn 6540 df-f 6541 |
This theorem is referenced by: usgrprc 29031 rgrusgrprc 29355 |
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