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Theorem epse 5531
Description: The membership relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the membership 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 5462 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
21bicomi 225 . . . . . 6 (𝑦𝑥𝑦 E 𝑥)
32abbi2i 2950 . . . . 5 𝑥 = {𝑦𝑦 E 𝑥}
4 vex 3495 . . . . 5 𝑥 ∈ V
53, 4eqeltrri 2907 . . . 4 {𝑦𝑦 E 𝑥} ∈ V
6 rabssab 4057 . . . 4 {𝑦𝐴𝑦 E 𝑥} ⊆ {𝑦𝑦 E 𝑥}
75, 6ssexi 5217 . . 3 {𝑦𝐴𝑦 E 𝑥} ∈ V
87rgenw 3147 . 2 𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V
9 df-se 5508 . 2 ( E Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V)
108, 9mpbir 232 1 E Se 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  {cab 2796  wral 3135  {crab 3139  Vcvv 3492   class class class wbr 5057   E cep 5457   Se wse 5505
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-ral 3140  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  df-se 5508
This theorem is referenced by:  omsinds  7589  tfr1ALT  8025  tfr2ALT  8026  tfr3ALT  8027  oieu  8991  oismo  8992  oiid  8993  cantnfp1lem3  9131  r0weon  9426  hsmexlem1  9836
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