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Mirrors > Home > MPE Home > Th. List > inex2 | Structured version Visualization version GIF version |
Description: Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.) |
Ref | Expression |
---|---|
inex2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
inex2 | ⊢ (𝐵 ∩ 𝐴) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4175 | . 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
2 | inex2.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 2 | inex1 5212 | . 2 ⊢ (𝐴 ∩ 𝐵) ∈ V |
4 | 1, 3 | eqeltri 2906 | 1 ⊢ (𝐵 ∩ 𝐴) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 |
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 |
This theorem is referenced by: ssex 5216 wefrc 5542 hartogslem1 8994 infxpenlem 9427 dfac5lem5 9541 fin23lem12 9741 fpwwe2lem12 10051 cnso 15588 ressbas 16542 ressress 16550 rescabs 17091 mgpress 19179 pjfval 20778 tgdom 21514 distop 21531 ustfilxp 22748 elovolmlem 24002 dyadmbl 24128 volsup2 24133 vitali 24141 itg1climres 24242 tayl0 24877 atomli 30086 ldgenpisyslem1 31321 reprinfz1 31792 bj-elid4 34352 aomclem6 39537 elinintrab 39815 isotone2 40277 ntrrn 40350 ntrf 40351 dssmapntrcls 40356 onfrALTlem3 40755 limcresiooub 41799 limcresioolb 41800 limsupval4 41951 sge0iunmptlemre 42574 ovolval2lem 42802 ovolval4lem2 42809 setrec2fun 44723 |
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