Theorem uhgredgss 29094

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 29091	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ (Edg‘𝐺)) → 𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
2	1	ex 412	. 2 ⊢ (𝐺 ∈ UHGraph → (𝑥 ∈ (Edg‘𝐺) → 𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})))
3	2	ssrdv 3943	1 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Syntax hints:   → wi 4   ∈ wcel 2109   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4553  {csn 4579  ‘cfv 6486  Vtxcvtx 28959  Edgcedg 29010  UHGraphcuhgr 29019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-edg 29011  df-uhgr 29021

This theorem is referenced by:  lfuhgr  35090