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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdinex1g | GIF version |
Description: Bounded version of inex1g 4059. (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 3265 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵)) | |
2 | 1 | eleq1d 2206 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V)) |
3 | bdinex1g.bd | . . 3 ⊢ BOUNDED 𝐵 | |
4 | vex 2684 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | bdinex1 13086 | . 2 ⊢ (𝑥 ∩ 𝐵) ∈ V |
6 | 2, 5 | vtoclg 2741 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∩ cin 3065 BOUNDED wbdc 13027 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-bdsep 13071 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-bdc 13028 |
This theorem is referenced by: (None) |
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