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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdinex1g | GIF version |
Description: Bounded version of inex1g 4100. (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 3301 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵)) | |
2 | 1 | eleq1d 2226 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V)) |
3 | bdinex1g.bd | . . 3 ⊢ BOUNDED 𝐵 | |
4 | vex 2715 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | bdinex1 13474 | . 2 ⊢ (𝑥 ∩ 𝐵) ∈ V |
6 | 2, 5 | vtoclg 2772 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 Vcvv 2712 ∩ cin 3101 BOUNDED wbdc 13415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-bdsep 13459 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-in 3108 df-bdc 13416 |
This theorem is referenced by: (None) |
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