| 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 2859 | 1 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ‘cfv 6561 Vtxcvtx 29013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: upgr1e 29130 uspgr1e 29261 nbgrval 29353 cplgr1vlem 29446 vtxdgval 29486 vtxdgelxnn0 29490 wlkson 29674 trlsonfval 29724 pthsonfval 29760 spthson 29761 2wlkd 29956 is0wlk 30136 0wlkon 30139 is0trl 30142 0trlon 30143 0pthon 30146 0clwlkv 30150 1wlkd 30160 3wlkd 30189 wlkl0 30386 clnbgrval 47809 isgrtri 47910 |
| Copyright terms: Public domain | W3C validator |