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Theorem epse 4272
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 4222 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
21bicomi 131 . . . . . 6 (𝑦𝑥𝑦 E 𝑥)
32abbi2i 2255 . . . . 5 𝑥 = {𝑦𝑦 E 𝑥}
4 vex 2692 . . . . 5 𝑥 ∈ V
53, 4eqeltrri 2214 . . . 4 {𝑦𝑦 E 𝑥} ∈ V
6 rabssab 3189 . . . 4 {𝑦𝐴𝑦 E 𝑥} ⊆ {𝑦𝑦 E 𝑥}
75, 6ssexi 4074 . . 3 {𝑦𝐴𝑦 E 𝑥} ∈ V
87rgenw 2490 . 2 𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V
9 df-se 4263 . 2 ( E Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦 E 𝑥} ∈ V)
108, 9mpbir 145 1 E Se 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1481  {cab 2126  wral 2417  {crab 2421  Vcvv 2689   class class class wbr 3937   E cep 4217   Se wse 4259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-eprel 4219  df-se 4263
This theorem is referenced by: (None)
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