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Theorem epse 5233
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 5166 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
21bicomi 214 . . . . . 6 (𝑦𝑥𝑦 E 𝑥)
32abbi2i 2887 . . . . 5 𝑥 = {𝑦𝑦 E 𝑥}
4 vex 3354 . . . . 5 𝑥 ∈ V
53, 4eqeltrri 2847 . . . 4 {𝑦𝑦 E 𝑥} ∈ V
6 rabssab 3841 . . . 4 {𝑦𝐴𝑦 E 𝑥} ⊆ {𝑦𝑦 E 𝑥}
75, 6ssexi 4938 . . 3 {𝑦𝐴𝑦 E 𝑥} ∈ V
87rgenw 3073 . 2 𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V
9 df-se 5210 . 2 ( E Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V)
108, 9mpbir 221 1 E Se 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  {cab 2757  wral 3061  {crab 3065  Vcvv 3351   class class class wbr 4787   E cep 5162   Se wse 5207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-eprel 5163  df-se 5210
This theorem is referenced by:  omsinds  7232  tfr1ALT  7650  tfr2ALT  7651  tfr3ALT  7652  oieu  8601  oismo  8602  oiid  8603  cantnfp1lem3  8742  r0weon  9036  hsmexlem1  9451
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