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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 5228 | . 2 ⊢ 𝐵 ∈ V |
4 | 3 | rabex 5226 | 1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 |
This theorem is referenced by: gsumbagdiag 20084 psrlidm 20111 psrridm 20112 psrass1 20113 mdegmullem 24599 vtxdginducedm1lem4 27251 vtxdginducedm1 27252 |
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