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Theorem epse 5295
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 5228 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
21bicomi 216 . . . . . 6 (𝑦𝑥𝑦 E 𝑥)
32abbi2i 2915 . . . . 5 𝑥 = {𝑦𝑦 E 𝑥}
4 vex 3388 . . . . 5 𝑥 ∈ V
53, 4eqeltrri 2875 . . . 4 {𝑦𝑦 E 𝑥} ∈ V
6 rabssab 3887 . . . 4 {𝑦𝐴𝑦 E 𝑥} ⊆ {𝑦𝑦 E 𝑥}
75, 6ssexi 4998 . . 3 {𝑦𝐴𝑦 E 𝑥} ∈ V
87rgenw 3105 . 2 𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V
9 df-se 5272 . 2 ( E Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V)
108, 9mpbir 223 1 E Se 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  {cab 2785  wral 3089  {crab 3093  Vcvv 3385   class class class wbr 4843   E cep 5224   Se wse 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-eprel 5225  df-se 5272
This theorem is referenced by:  omsinds  7318  tfr1ALT  7735  tfr2ALT  7736  tfr3ALT  7737  oieu  8686  oismo  8687  oiid  8688  cantnfp1lem3  8827  r0weon  9121  hsmexlem1  9536
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