Theorem subgrv 27055

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 27054	. 2 ⊢ Rel SubGraph
2	1	brrelex12i 5610	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2113  Vcvv 3497   class class class wbr 5069   SubGraph csubgr 27052

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-subgr 27053

This theorem is referenced by:  subgrprop  27058  subgrprop3  27061  subuhgr  27071  subupgr  27072  subumgr  27073  subusgr  27074  subgrwlk  32383  acycgrsubgr  32409