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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdinex1g | GIF version |
Description: Bounded version of inex1g 4151. (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 3341 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵)) | |
2 | 1 | eleq1d 2256 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V)) |
3 | bdinex1g.bd | . . 3 ⊢ BOUNDED 𝐵 | |
4 | vex 2752 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | bdinex1 14891 | . 2 ⊢ (𝑥 ∩ 𝐵) ∈ V |
6 | 2, 5 | vtoclg 2809 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 Vcvv 2749 ∩ cin 3140 BOUNDED wbdc 14832 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-bdsep 14876 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-in 3147 df-bdc 14833 |
This theorem is referenced by: (None) |
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