Theorem subgrv 16106

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 e. _V /\ G e. _V ) )

Proof of Theorem subgrv

Step	Hyp	Ref	Expression
1		relsubgr 16105	. 2 |- Rel SubGraph
2	1	brrelex12i 4768	1 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   SubGraph csubgr 16103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-subgr 16104

This theorem is referenced by:  subgrprop  16109  subgrprop3  16112  subuhgr  16122  subupgr  16123  subumgr  16124  subusgr  16125