Theorem subgrv 26745

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 26744	. 2 ⊢ Rel SubGraph
2	1	brrelex12i 5450	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))

Syntax hints:   → wi 4   ∧ wa 387   ∈ wcel 2048  Vcvv 3409   class class class wbr 4923   SubGraph csubgr 26742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-xp 5406  df-rel 5407  df-subgr 26743

This theorem is referenced by:  subgrprop  26748  subgrprop3  26751  subuhgr  26761  subupgr  26762  subumgr  26763  subusgr  26764