Theorem subgrv 29355

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 29354	. 2 ⊢ Rel SubGraph
2	1	brrelex12i 5687	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3442   class class class wbr 5100   SubGraph csubgr 29352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-subgr 29353

This theorem is referenced by:  subgrprop  29358  subgrprop3  29361  subuhgr  29371  subupgr  29372  subumgr  29373  subusgr  29374  subgrwlk  35345  acycgrsubgr  35371