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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. (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 3495 | . 2 ⊢ 𝑥 ∈ V | |
2 | 1 | epeli 5461 | 1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2105 class class class wbr 5057 E cep 5457 |
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 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-eprel 5458 |
This theorem is referenced by: epse 5531 dfepfr 5533 epfrc 5534 wecmpep 5540 wetrep 5541 dmep 5786 domepOLD 5787 rnep 5790 epweon 7486 smoiso 7988 smoiso2 7995 ordunifi 8756 ordiso2 8967 ordtypelem8 8977 oismo 8992 wofib 8997 dford2 9071 noinfep 9111 oemapso 9133 wemapwe 9148 alephiso 9512 cflim2 9673 fin23lem27 9738 om2uzisoi 13310 bnj219 31902 efrunt 32836 dftr6 32883 dffr5 32886 elpotr 32923 dfon2lem9 32933 dfon2 32934 brsset 33247 dfon3 33250 brbigcup 33256 brapply 33296 brcup 33297 brcap 33298 dfint3 33310 dfssr2 35619 |
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