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| Mirrors > Home > MPE Home > Th. List > epel | Structured version Visualization version GIF version | ||
| Description: The membership relation and the membership predicate agree when the "containing" class is a setvar. Definition 1.6 of [Schloeder] p. 1. (Contributed by NM, 13-Aug-1995.) Replace the first setvar variable with a class variable. (Revised by BJ, 13-Sep-2022.) |
| Ref | Expression |
|---|---|
| epel | ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . 2 ⊢ 𝑥 ∈ V | |
| 2 | 1 | epeli 5586 | 1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 class class class wbr 5143 E cep 5583 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 |
| This theorem is referenced by: epse 5667 dfepfr 5669 epfrc 5670 wecmpep 5677 wetrep 5678 dmep 5934 rnep 5937 epweon 7795 epweonALT 7796 smoiso 8402 smoiso2 8409 ordunifi 9326 ordiso2 9555 ordtypelem8 9565 oismo 9580 wofib 9585 dford2 9660 noinfep 9700 oemapso 9722 wemapwe 9737 alephiso 10138 cflim2 10303 fin23lem27 10368 om2uzisoi 13995 om2noseqiso 28308 bnj219 34747 nummin 35105 efrunt 35713 dftr6 35751 dffr5 35754 elpotr 35782 dfon2lem9 35792 dfon2 35793 brsset 35890 dfon3 35893 brbigcup 35899 brapply 35939 brcup 35940 brcap 35941 dfint3 35953 dfssr2 38500 onsupuni 43241 onsupmaxb 43251 rankrelp 44977 sswfaxreg 45004 |
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