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Theorem subgrv 27058
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
StepHypRef Expression
1 relsubgr 27057 . 2 Rel SubGraph
21brrelex12i 5584 1 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  Vcvv 3469   class class class wbr 5042   SubGraph csubgr 27055
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-subgr 27056
This theorem is referenced by:  subgrprop  27061  subgrprop3  27064  subuhgr  27074  subupgr  27075  subumgr  27076  subusgr  27077  subgrwlk  32453  acycgrsubgr  32479
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