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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 31763 disjdifprg2 31807 sigainb 33134 ldgenpisyslem1 33161 carsgclctun 33320 eulerpartlemgs2 33379 sseqval 33387 reprinrn 33630 bnj1177 34017 cvmsss2 34265 satef 34407 satefvfmla0 34409 fnemeet2 35252 ontgval 35316 bj-discrmoore 35992 bj-ideqb 36040 bj-opelidres 36042 bj-opelidb1ALT 36047 fin2so 36475 inex3 37207 inxpex 37208 dfrefrels2 37383 dfsymrels2 37415 dftrrels2 37445 elrfi 41432 ofoafg 42104 fourierdlem71 44893 fourierdlem80 44902 sge0less 45108 sge0ssre 45113 carageniuncllem2 45238 dfrngc2 46870 rnghmsscmap2 46871 rngcbasALTV 46881 dfringc2 46916 rhmsscmap2 46917 rhmsscrnghm 46924 rngcresringcat 46928 ringcbasALTV 46944 srhmsubc 46974 fldc 46981 fldhmsubc 46982 rngcrescrhm 46983 srhmsubcALTV 46992 fldcALTV 46999 fldhmsubcALTV 47000 rngcrescrhmALTV 47001 |
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