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Mirrors > Home > MPE Home > Th. List > uhgriedg0edg0 | Structured version Visualization version GIF version |
Description: A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 8-Dec-2021.) |
Ref | Expression |
---|---|
uhgriedg0edg0 | ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
2 | 1 | uhgrfun 26851 | . 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
3 | eqid 2821 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 3 | edg0iedg0 26840 | . 2 ⊢ (Fun (iEdg‘𝐺) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
5 | 2, 4 | syl 17 | 1 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∅c0 4291 Fun wfun 6349 ‘cfv 6355 iEdgciedg 26782 Edgcedg 26832 UHGraphcuhgr 26841 |
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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-edg 26833 df-uhgr 26843 |
This theorem is referenced by: uhgr0v0e 27020 uhgr0vusgr 27024 lfuhgr1v0e 27036 usgr1vr 27037 usgr1v0e 27108 uhgr0edg0rgr 27355 rgrusgrprc 27371 |
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