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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 6705 | . 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V) | |
2 | 1vgrex.v | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 1, 2 | eleq2s 2933 | 1 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ‘cfv 6357 Vtxcvtx 26783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-dm 5567 df-iota 6316 df-fv 6365 |
This theorem is referenced by: upgr1e 26900 uspgr1e 27028 nbgrval 27120 cplgr1vlem 27213 vtxdgval 27252 vtxdgelxnn0 27256 wlkson 27440 trlsonfval 27489 pthsonfval 27523 spthson 27524 2wlkd 27717 is0wlk 27898 0wlkon 27901 is0trl 27904 0trlon 27905 0pthon 27908 0clwlkv 27912 1wlkd 27922 3wlkd 27951 wlkl0 28148 |
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