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Mirrors > Home > MPE Home > Th. List > rab2ex | Structured version Visualization version GIF version |
Description: A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.) |
Ref | Expression |
---|---|
rab2ex.1 | ⊢ 𝐵 = {𝑦 ∈ 𝐴 ∣ 𝜓} |
rab2ex.2 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rab2ex | ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rab2ex.1 | . . 3 ⊢ 𝐵 = {𝑦 ∈ 𝐴 ∣ 𝜓} | |
2 | rab2ex.2 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | rabex2 5336 | . 2 ⊢ 𝐵 ∈ V |
4 | 3 | rabex 5334 | 1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 {crab 3429 Vcvv 3471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-in 3954 df-ss 3964 |
This theorem is referenced by: gsumbagdiagOLD 21873 gsumbagdiag 21876 psrlidm 21905 psrridm 21906 psrass1 21907 mdegmullem 26027 vtxdginducedm1lem4 29369 vtxdginducedm1 29370 |
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