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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdrabexg | GIF version |
Description: Bounded version of rabexg 3891. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdrabexg.bd | ⊢ BOUNDED φ |
bdrabexg.bdc | ⊢ BOUNDED A |
Ref | Expression |
---|---|
bdrabexg | ⊢ (A ∈ 𝑉 → {x ∈ A ∣ φ} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3019 | . 2 ⊢ {x ∈ A ∣ φ} ⊆ A | |
2 | bdrabexg.bdc | . . . 4 ⊢ BOUNDED A | |
3 | bdrabexg.bd | . . . 4 ⊢ BOUNDED φ | |
4 | 2, 3 | bdcrab 9307 | . . 3 ⊢ BOUNDED {x ∈ A ∣ φ} |
5 | 4 | bdssexg 9359 | . 2 ⊢ (({x ∈ A ∣ φ} ⊆ A ∧ A ∈ 𝑉) → {x ∈ A ∣ φ} ∈ V) |
6 | 1, 5 | mpan 400 | 1 ⊢ (A ∈ 𝑉 → {x ∈ A ∣ φ} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1390 {crab 2304 Vcvv 2551 ⊆ wss 2911 BOUNDED wbd 9267 BOUNDED wbdc 9295 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-bd0 9268 ax-bdan 9270 ax-bdsb 9277 ax-bdsep 9339 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 df-v 2553 df-in 2918 df-ss 2925 df-bdc 9296 |
This theorem is referenced by: bj-inex 9362 |
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