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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 5334 | . 2 ⊢ 𝐵 ∈ V |
4 | 3 | rabex 5332 | 1 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-in 3955 df-ss 3965 |
This theorem is referenced by: gsumbagdiagOLD 21802 gsumbagdiag 21805 psrlidm 21834 psrridm 21835 psrass1 21836 mdegmullem 25934 vtxdginducedm1lem4 29232 vtxdginducedm1 29233 |
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