![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 1vgrex | Structured version Visualization version GIF version |
Description: A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.) |
Ref | Expression |
---|---|
1vgrex.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
1vgrex | ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6944 | . 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V) | |
2 | 1vgrex.v | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 1, 2 | eleq2s 2856 | 1 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ‘cfv 6562 Vtxcvtx 29027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-dm 5698 df-iota 6515 df-fv 6570 |
This theorem is referenced by: upgr1e 29144 uspgr1e 29275 nbgrval 29367 cplgr1vlem 29460 vtxdgval 29500 vtxdgelxnn0 29504 wlkson 29688 trlsonfval 29738 pthsonfval 29772 spthson 29773 2wlkd 29965 is0wlk 30145 0wlkon 30148 is0trl 30151 0trlon 30152 0pthon 30155 0clwlkv 30159 1wlkd 30169 3wlkd 30198 wlkl0 30395 clnbgrval 47746 isgrtri 47847 |
Copyright terms: Public domain | W3C validator |