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Theorem uhgredgrnv 29148
Description: An edge of a hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 4-Jun-2021.)
Assertion
Ref Expression
uhgredgrnv ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑁𝐸) → 𝑁 ∈ (Vtx‘𝐺))

Proof of Theorem uhgredgrnv
StepHypRef Expression
1 edguhgr 29147 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ 𝒫 (Vtx‘𝐺))
2 elelpwi 4609 . . . 4 ((𝑁𝐸𝐸 ∈ 𝒫 (Vtx‘𝐺)) → 𝑁 ∈ (Vtx‘𝐺))
32expcom 413 . . 3 (𝐸 ∈ 𝒫 (Vtx‘𝐺) → (𝑁𝐸𝑁 ∈ (Vtx‘𝐺)))
41, 3syl 17 . 2 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝑁𝐸𝑁 ∈ (Vtx‘𝐺)))
543impia 1117 1 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑁𝐸) → 𝑁 ∈ (Vtx‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2107  𝒫 cpw 4599  cfv 6560  Vtxcvtx 29014  Edgcedg 29065  UHGraphcuhgr 29074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-edg 29066  df-uhgr 29076
This theorem is referenced by:  clnbgredg  47831  usgrgrtrirex  47922
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