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| Mirrors > Home > ILE Home > Th. List > Mathboxes > bdinex1g | GIF version | ||
| Description: Bounded version of inex1g 4179. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bdinex1g.bd | ⊢ BOUNDED 𝐵 |
| Ref | Expression |
|---|---|
| bdinex1g | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ineq1 3366 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵)) | |
| 2 | 1 | eleq1d 2273 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V)) |
| 3 | bdinex1g.bd | . . 3 ⊢ BOUNDED 𝐵 | |
| 4 | vex 2774 | . . 3 ⊢ 𝑥 ∈ V | |
| 5 | 3, 4 | bdinex1 15797 | . 2 ⊢ (𝑥 ∩ 𝐵) ∈ V |
| 6 | 2, 5 | vtoclg 2832 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 = wceq 1372 ∈ wcel 2175 Vcvv 2771 ∩ cin 3164 BOUNDED wbdc 15738 |
| 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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 ax-bdsep 15782 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-v 2773 df-in 3171 df-bdc 15739 |
| This theorem is referenced by: (None) |
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