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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdrabexg | GIF version |
Description: Bounded version of rabexg 4172. (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 3264 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
2 | bdrabexg.bdc | . . . 4 ⊢ BOUNDED 𝐴 | |
3 | bdrabexg.bd | . . . 4 ⊢ BOUNDED 𝜑 | |
4 | 2, 3 | bdcrab 15344 | . . 3 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑} |
5 | 4 | bdssexg 15396 | . 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
6 | 1, 5 | mpan 424 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 {crab 2476 Vcvv 2760 ⊆ wss 3153 BOUNDED wbd 15304 BOUNDED wbdc 15332 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 ax-bd0 15305 ax-bdan 15307 ax-bdsb 15314 ax-bdsep 15376 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-rab 2481 df-v 2762 df-in 3159 df-ss 3166 df-bdc 15333 |
This theorem is referenced by: bj-inex 15399 |
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