Theorem uhgredgss 29204

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 29201	. . 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 3939	1 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Syntax hints:   → wi 4   ∈ wcel 2113   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ‘cfv 6492  Vtxcvtx 29069  Edgcedg 29120  UHGraphcuhgr 29129

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-edg 29121  df-uhgr 29131

This theorem is referenced by:  lfuhgr  35312