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Theorem epse 5126
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 5061 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
21bicomi 214 . . . . . 6 (𝑦𝑥𝑦 E 𝑥)
32abbi2i 2767 . . . . 5 𝑥 = {𝑦𝑦 E 𝑥}
4 vex 3234 . . . . 5 𝑥 ∈ V
53, 4eqeltrri 2727 . . . 4 {𝑦𝑦 E 𝑥} ∈ V
6 rabssab 3723 . . . 4 {𝑦𝐴𝑦 E 𝑥} ⊆ {𝑦𝑦 E 𝑥}
75, 6ssexi 4836 . . 3 {𝑦𝐴𝑦 E 𝑥} ∈ V
87rgenw 2953 . 2 𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V
9 df-se 5103 . 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 2030  {cab 2637  wral 2941  {crab 2945  Vcvv 3231   class class class wbr 4685   E cep 5057   Se wse 5100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-eprel 5058  df-se 5103
This theorem is referenced by:  omsinds  7126  tfr1ALT  7541  tfr2ALT  7542  tfr3ALT  7543  oieu  8485  oismo  8486  oiid  8487  cantnfp1lem3  8615  r0weon  8873  hsmexlem1  9286
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