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Theorem epse 4079
 Description: The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
epse E Se 𝐴

Proof of Theorem epse
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epel 4029 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
21bicomi 123 . . . . . 6 (𝑦𝑥𝑦 E 𝑥)
32abbi2i 2152 . . . . 5 𝑥 = {𝑦𝑦 E 𝑥}
4 vex 2560 . . . . 5 𝑥 ∈ V
53, 4eqeltrri 2111 . . . 4 {𝑦𝑦 E 𝑥} ∈ V
6 rabssab 3027 . . . 4 {𝑦𝐴𝑦 E 𝑥} ⊆ {𝑦𝑦 E 𝑥}
75, 6ssexi 3895 . . 3 {𝑦𝐴𝑦 E 𝑥} ∈ V
87rgenw 2376 . 2 𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V
9 df-se 4070 . 2 ( E Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V)
108, 9mpbir 134 1 E Se 𝐴
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  {cab 2026  ∀wral 2306  {crab 2310  Vcvv 2557   class class class wbr 3764   E cep 4024   Se wse 4066 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026  df-se 4070 This theorem is referenced by: (None)
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