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Mirrors > Home > MPE Home > Th. List > inex1g | Structured version Visualization version GIF version |
Description: Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
Ref | Expression |
---|---|
inex1g | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 4206 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵)) | |
2 | 1 | eleq1d 2819 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V)) |
3 | vex 3479 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | inex1 5318 | . 2 ⊢ (𝑥 ∩ 𝐵) ∈ V |
5 | 2, 4 | vtoclg 3557 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cin 3948 |
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 |
This theorem is referenced by: inex2g 5321 dmresexg 6006 predexg 6319 onin 6396 offval 7679 offval3 7969 frrlem13 8283 onsdominel 9126 ssenen 9151 inelfi 9413 fiin 9417 tskwe 9945 dfac8b 10026 ac10ct 10029 infpwfien 10057 fictb 10240 canthnum 10644 gruina 10813 ressinbas 17190 ressress 17193 qusin 17490 catcbas 18051 fpwipodrs 18493 psss 18533 gsumzres 19777 eltg 22460 eltg3 22465 ntrval 22540 restco 22668 restfpw 22683 ordtrest 22706 ordtrest2lem 22707 ordtrest2 22708 cnrmi 22864 restcnrm 22866 kgeni 23041 tsmsfbas 23632 eltsms 23637 tsmsres 23648 caussi 24814 causs 24815 elpwincl1 31794 disjdifprg2 31838 sigainb 33165 ldgenpisyslem1 33192 carsgclctun 33351 eulerpartlemgs2 33410 sseqval 33418 reprinrn 33661 bnj1177 34048 cvmsss2 34296 satef 34438 satefvfmla0 34440 fnemeet2 35300 ontgval 35364 bj-discrmoore 36040 bj-ideqb 36088 bj-opelidres 36090 bj-opelidb1ALT 36095 fin2so 36523 inex3 37255 inxpex 37256 dfrefrels2 37431 dfsymrels2 37463 dftrrels2 37493 elrfi 41480 ofoafg 42152 fourierdlem71 44941 fourierdlem80 44950 sge0less 45156 sge0ssre 45161 carageniuncllem2 45286 dfrngc2 46918 rnghmsscmap2 46919 rngcbasALTV 46929 dfringc2 46964 rhmsscmap2 46965 rhmsscrnghm 46972 rngcresringcat 46976 ringcbasALTV 46992 srhmsubc 47022 fldc 47029 fldhmsubc 47030 rngcrescrhm 47031 srhmsubcALTV 47040 fldcALTV 47047 fldhmsubcALTV 47048 rngcrescrhmALTV 47049 |
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