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Mirrors > Home > MPE Home > Th. List > rabexg | Structured version Visualization version GIF version |
Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) (Proof shortened by BJ, 24-Jul-2025.) |
Ref | Expression |
---|---|
rabexg | ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabelpw 5341 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴) | |
2 | 1 | elexd 3501 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 {crab 3432 Vcvv 3477 𝒫 cpw 4604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-in 3969 df-ss 3979 df-pw 4606 |
This theorem is referenced by: rabex 5344 rabexd 5345 class2set 5360 exse 5648 elfvmptrab1w 7042 elfvmptrab1 7043 elovmporab 7678 elovmporab1w 7679 elovmporab1 7680 ovmpt3rabdm 7691 elovmpt3rab1 7692 suppval 8185 mpoxopoveq 8242 wdom2d 9617 scottex 9922 tskwe 9987 fin1a2lem12 10448 hashbclem 14487 wrdnfi 14582 wrd2f1tovbij 14995 hashdvds 16808 hashbcval 17035 brric 20520 psrass1lem 21969 psrcom 22005 dmatval 22513 cpmat 22730 fctop 23026 cctop 23028 ppttop 23029 epttop 23031 cldval 23046 neif 23123 neival 23125 neiptoptop 23154 neiptopnei 23155 ordtbaslem 23211 ordtbas2 23214 ordtopn1 23217 ordtopn2 23218 ordtrest2lem 23226 cmpsublem 23422 kgenval 23558 qtopval 23718 kqfval 23746 ordthmeolem 23824 elmptrab 23850 fbssfi 23860 fgval 23893 flimval 23986 flimfnfcls 24051 ptcmplem2 24076 ptcmplem3 24077 tsmsfbas 24151 eltsms 24156 utopval 24256 blvalps 24410 blval 24411 minveclem3b 25475 minveclem3 25476 minveclem4 25479 cutlt 27980 fusgredgfi 29356 nbgrval 29367 cusgrsize 29486 wwlks 29864 wwlksnextbij 29931 clwwlk 30011 vdn0conngrumgrv2 30224 vdgn1frgrv2 30324 frgrwopreglem1 30340 rabfodom 32532 ordtrest2NEWlem 33882 hasheuni 34065 sigaval 34091 ldgenpisyslem1 34143 ddemeas 34216 braew 34222 imambfm 34243 carsgval 34284 iscvm 35243 cvmsval 35250 fwddifval 36143 fnessref 36339 indexa 37719 supex2g 37723 rfovfvfvd 43992 rfovcnvf1od 43993 fsovfvfvd 44000 fsovcnvlem 44002 cnfex 44965 stoweidlem26 45981 stoweidlem31 45986 stoweidlem34 45989 stoweidlem46 46001 stoweidlem59 46014 salexct 46289 caragenval 46448 clnbgrval 47746 dmatALTbas 48246 lcoop 48256 |
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