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Mirrors > Home > ILE Home > Th. List > elexd | GIF version |
Description: If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
elexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
elexd | ⊢ (𝜑 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | elex 2630 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 Vcvv 2619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-v 2621 |
This theorem is referenced by: tfr1onlemsucfn 6105 tfrcllemsucfn 6118 frecrdg 6173 unsnfidcel 6631 fnfi 6646 seq3val 9874 hashennn 10188 lcmval 11323 hashdvds 11475 isstruct2r 11505 strnfvnd 11515 strfvssn 11518 strslfv2d 11536 setsslid 11544 ressid2 11552 ressval2 11553 istopon 11610 |
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