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Mirrors > Home > MPE Home > Th. List > vtxvalprc | Structured version Visualization version GIF version |
Description: Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.) |
Ref | Expression |
---|---|
vtxvalprc | ⊢ (𝐶 ∉ V → (Vtx‘𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3124 | . 2 ⊢ (𝐶 ∉ V ↔ ¬ 𝐶 ∈ V) | |
2 | fvprc 6663 | . 2 ⊢ (¬ 𝐶 ∈ V → (Vtx‘𝐶) = ∅) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝐶 ∉ V → (Vtx‘𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 Vcvv 3494 ∅c0 4291 ‘cfv 6355 Vtxcvtx 26781 |
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 2793 ax-nul 5210 ax-pow 5266 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 |
This theorem is referenced by: wlk0prc 27435 |
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