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Mirrors > Home > ILE Home > Th. List > elex | Unicode version |
Description: If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
elex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exsimpl 1508 | . 2 | |
2 | df-clel 2036 | . 2 | |
3 | isset 2561 | . 2 | |
4 | 1, 2, 3 | 3imtr4i 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: elexi 2567 elisset 2568 vtoclgft 2604 vtoclgf 2612 vtocl2gf 2615 vtocl3gf 2616 spcimgft 2629 spcimegft 2631 elab4g 2691 elrabf 2696 mob 2723 sbcex 2772 sbcabel 2839 csbcomg 2873 csbvarg 2877 csbiebt 2886 csbnestgf 2898 csbidmg 2902 sbcco3g 2903 csbco3g 2904 eldif 2927 ssv 2965 elun 3084 elin 3126 snidb 3401 dfopg 3547 eluni 3583 eliun 3661 csbexga 3885 nvel 3889 class2seteq 3916 axpweq 3924 snelpwi 3948 prelpwi 3950 opexg 3964 elopab 3995 epelg 4027 elon2 4113 unexg 4178 reuhypd 4203 op1stbg 4210 sucexg 4224 sucelon 4229 onsucelsucr 4234 sucunielr 4236 en2lp 4278 peano2 4318 peano2b 4337 opelvvg 4389 opeliunxp 4395 opbrop 4419 opeliunxp2 4476 ideqg 4487 elrnmptg 4586 imasng 4690 iniseg 4697 opswapg 4807 elxp4 4808 elxp5 4809 dmmptg 4818 iota2 4893 fnmpt 5025 fvexg 5194 fvelimab 5229 mptfvex 5256 fvmptdf 5258 fvmptdv2 5260 mpteqb 5261 fvmptt 5262 fvmptf 5263 fvopab6 5264 fsn2 5337 fmptpr 5355 fliftel 5433 eloprabga 5591 ovmpt2s 5624 ov2gf 5625 ovmpt2dxf 5626 ovmpt2x 5629 ovmpt2ga 5630 ovmpt2df 5632 ovi3 5637 ovelrn 5649 suppssfv 5708 suppssov1 5709 offval3 5761 1stexg 5794 2ndexg 5795 elxp6 5796 elxp7 5797 releldm2 5811 fnmpt2 5828 mpt2fvex 5829 mpt2exg 5834 brtpos2 5866 rdgtfr 5961 rdgruledefgg 5962 frec0g 5983 sucinc2 6026 nntri3or 6072 relelec 6146 ecdmn0m 6148 elinp 6572 addnqprlemfl 6657 addnqprlemfu 6658 mulnqprlemfl 6673 mulnqprlemfu 6674 recexprlemell 6720 recexprlemelu 6721 shftfvalg 9419 clim 9802 climmpt 9821 climshft2 9827 bj-vtoclgft 9914 bj-nvel 10017 |
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