Theorem uhgredgss 29715

Description: The set of edges of a hypergraph is a subset of the power set of vertices without the empty set. (Contributed by AV, 29-Nov-2020.)

Assertion

Ref	Expression
uhgredgss	⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Proof of Theorem uhgredgss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgredgn0 29712	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
2	1	ex 418	. 2 ⊢ (𝐺 ∈ UHGraph → (𝑥 ∈ (Edg‘𝐺) → 𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})))
3	2	ssrdv 3914	1 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  {csn 4584  ‘cfv 6539  Vtxcvtx 29580  Edgcedg 29631  UHGraphcuhgr 29640

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-edg 29632  df-uhgr 29642

This theorem is used by:  lfuhgr  29732