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Mirrors > Home > MPE Home > Th. List > rabex2 | Structured version Visualization version GIF version |
Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.) |
Ref | Expression |
---|---|
rabex2.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
rabex2.2 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rabex2 | ⊢ 𝐵 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabex2.2 | . 2 ⊢ 𝐴 ∈ V | |
2 | rabex2.1 | . . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} | |
3 | id 22 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
4 | 2, 3 | rabexd 5334 | . 2 ⊢ (𝐴 ∈ V → 𝐵 ∈ V) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ 𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-in 3956 df-ss 3966 |
This theorem is referenced by: rab2ex 5336 mapfien2 9404 cantnffval 9658 nqex 10918 gsumvalx 18595 psgnfval 19368 odval 19402 sylow1lem2 19467 sylow3lem6 19500 ablfaclem1 19955 psrass1lemOLD 21493 psrass1lem 21496 psrbas 21497 psrelbas 21498 psrmulfval 21504 psrmulcllem 21506 psrvscaval 21511 psr0cl 21513 psr0lid 21514 psrnegcl 21515 psrlinv 21516 psr1cl 21522 psrlidm 21523 psrdi 21526 psrdir 21527 psrass23l 21528 psrcom 21529 psrass23 21530 mvrval 21541 mplsubglem 21558 mpllsslem 21559 mplsubrglem 21563 mplvscaval 21575 mplmon 21590 mplmonmul 21591 mplcoe1 21592 ltbval 21598 mplmon2 21622 evlslem2 21642 evlslem3 21643 evlslem1 21645 rrxmet 24925 mdegldg 25584 lgamgulmlem5 26537 lgamgulmlem6 26538 lgamgulm2 26540 lgamcvglem 26544 upgrres1lem1 28597 frgrwopreg1 29602 dlwwlknondlwlknonen 29650 nsgmgc 32554 nsgqusf1o 32558 eulerpartlem1 33397 eulerpartlemt 33401 eulerpartgbij 33402 ballotlemoex 33515 satffunlem2lem2 34428 mapdunirnN 40569 mplmapghm 41176 evlsvvvallem2 41182 selvvvval 41205 evlsmhpvvval 41215 pwfi2en 41887 smfresal 45552 oddiadd 46632 2zrngadd 46883 2zrngmul 46891 |
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