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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 6678 | . 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → 𝐺 ∈ V) | |
2 | 1vgrex.v | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 1, 2 | eleq2s 2908 | 1 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ‘cfv 6324 Vtxcvtx 26789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 ax-pow 5231 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 |
This theorem is referenced by: upgr1e 26906 uspgr1e 27034 nbgrval 27126 cplgr1vlem 27219 vtxdgval 27258 vtxdgelxnn0 27262 wlkson 27446 trlsonfval 27495 pthsonfval 27529 spthson 27530 2wlkd 27722 is0wlk 27902 0wlkon 27905 is0trl 27908 0trlon 27909 0pthon 27912 0clwlkv 27916 1wlkd 27926 3wlkd 27955 wlkl0 28152 |
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