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Theorem epse 5518
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 5447 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
21bicomi 227 . . . . . 6 (𝑦𝑥𝑦 E 𝑥)
32abbi2i 2872 . . . . 5 𝑥 = {𝑦𝑦 E 𝑥}
4 vex 3404 . . . . 5 𝑥 ∈ V
53, 4eqeltrri 2831 . . . 4 {𝑦𝑦 E 𝑥} ∈ V
6 rabssab 3984 . . . 4 {𝑦𝐴𝑦 E 𝑥} ⊆ {𝑦𝑦 E 𝑥}
75, 6ssexi 5200 . . 3 {𝑦𝐴𝑦 E 𝑥} ∈ V
87rgenw 3066 . 2 𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V
9 df-se 5494 . 2 ( E Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V)
108, 9mpbir 234 1 E Se 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2114  {cab 2717  wral 3054  {crab 3058  Vcvv 3400   class class class wbr 5040   E cep 5443   Se wse 5491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-ral 3059  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-eprel 5444  df-se 5494
This theorem is referenced by:  omsinds  7632  tfr1ALT  8078  tfr2ALT  8079  tfr3ALT  8080  oieu  9089  oismo  9090  oiid  9091  cantnfp1lem3  9229  r0weon  9525  hsmexlem1  9939  on2recsfn  33482  on2recsov  33483  on2ind  33484  on3ind  33485
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