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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 5239 | . 2 ⊢ 𝐵 ∈ V |
4 | 3 | rabex 5237 | 1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-in 3945 df-ss 3954 |
This theorem is referenced by: gsumbagdiag 20158 psrlidm 20185 psrridm 20186 psrass1 20187 mdegmullem 24674 vtxdginducedm1lem4 27326 vtxdginducedm1 27327 |
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