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Mirrors > Home > MPE Home > Th. List > subgrv | Structured version Visualization version GIF version |
Description: If a class is a subgraph of another class, both classes are sets. (Contributed by AV, 16-Nov-2020.) |
Ref | Expression |
---|---|
subgrv | ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsubgr 27059 | . 2 ⊢ Rel SubGraph | |
2 | 1 | brrelex12i 5571 | 1 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 SubGraph csubgr 27057 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
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 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-subgr 27058 |
This theorem is referenced by: subgrprop 27063 subgrprop3 27066 subuhgr 27076 subupgr 27077 subumgr 27078 subusgr 27079 subgrwlk 32492 acycgrsubgr 32518 |
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