Theorem subgrv 29525

Description: If a class is a subgraph of another class, both classes are sets. (Contributed by AV, 16-Nov-2020.)

Assertion

Ref	Expression
subgrv	⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))

Proof of Theorem subgrv

Step	Hyp	Ref	Expression
1		relsubgr 29524	. 2 ⊢ Rel SubGraph
2	1	brrelex12i 5706	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  Vcvv 3457   class class class wbr 5104   SubGraph csubgr 29522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5657  df-rel 5658  df-subgr 29523

This theorem is referenced by:  subgrprop  29528  subgrprop3  29531  subuhgr  29541  subupgr  29542  subumgr  29543  subusgr  29544  subgrwlk  35490  acycgrsubgr  35516