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Mirrors > Home > ILE Home > Th. List > inex1g | GIF version |
Description: Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
Ref | Expression |
---|---|
inex1g | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 3131 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵)) | |
2 | 1 | eleq1d 2106 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V)) |
3 | vex 2560 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | inex1 3891 | . 2 ⊢ (𝑥 ∩ 𝐵) ∈ V |
5 | 2, 4 | vtoclg 2613 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∩ cin 2916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 |
This theorem is referenced by: onin 4123 dmresexg 4634 funimaexg 4983 offval 5719 offval3 5761 |
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