Theorem subgrv 27046

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 27045	. 2 ⊢ Rel SubGraph
2	1	brrelex12i 5601	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  Vcvv 3494   class class class wbr 5058   SubGraph csubgr 27043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-subgr 27044

This theorem is referenced by:  subgrprop  27049  subgrprop3  27052  subuhgr  27062  subupgr  27063  subumgr  27064  subusgr  27065  subgrwlk  32374  acycgrsubgr  32400