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Mirrors > Home > MPE Home > Th. List > iedgvalprc | Structured version Visualization version GIF version |
Description: Degenerated case 4 for edges: The set of indexed edges of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.) |
Ref | Expression |
---|---|
iedgvalprc | ⊢ (𝐶 ∉ V → (iEdg‘𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3124 | . 2 ⊢ (𝐶 ∉ V ↔ ¬ 𝐶 ∈ V) | |
2 | fvprc 6662 | . 2 ⊢ (¬ 𝐶 ∈ V → (iEdg‘𝐶) = ∅) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝐶 ∉ V → (iEdg‘𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ∉ wnel 3123 Vcvv 3494 ∅c0 4290 ‘cfv 6354 iEdgciedg 26781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5209 ax-pow 5265 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 |
This theorem is referenced by: (None) |
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