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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 3329 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵)) | |
2 | 1 | eleq1d 2246 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V)) |
3 | vex 2740 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | inex1 4136 | . 2 ⊢ (𝑥 ∩ 𝐵) ∈ V |
5 | 2, 4 | vtoclg 2797 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4120 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 |
This theorem is referenced by: onin 4385 dmresexg 4929 funimaexg 5299 offval 6087 offval3 6132 ssenen 6848 ressvalsets 12516 ressex 12517 ressbasd 12519 resseqnbasd 12524 ressinbasd 12525 ressressg 12526 mgpress 13072 isunitd 13206 eltg 13423 eltg3 13428 ntrval 13481 restco 13545 |
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