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Theorem subgrv 26143
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 26142 . . 3 Rel SubGraph
21brrelexi 5148 . 2 (𝑆 SubGraph 𝐺𝑆 ∈ V)
31brrelex2i 5149 . 2 (𝑆 SubGraph 𝐺𝐺 ∈ V)
42, 3jca 554 1 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  Vcvv 3195   class class class wbr 4644   SubGraph csubgr 26140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-subgr 26141
This theorem is referenced by:  subgrprop  26146  subgrprop3  26149  subuhgr  26159  subupgr  26160  subumgr  26161  subusgr  26162
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