Theorem epse 5660

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.

Step	Hyp	Ref	Expression
1		epel 5584	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
2	1	bicomi 223	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥)
3	2	eqabi 2870	. . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥}
4		vex 3479	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltrri 2831	. . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V
6		rabssab 4084	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥}
7	5, 6	ssexi 5323	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
8	7	rgenw 3066	. 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
9		df-se 5633	. 2 ⊢ ( E Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V)
10	8, 9	mpbir 230	1 ⊢ E Se 𝐴

Syntax hints:   ∈ wcel 2107  {cab 2710  ∀wral 3062  {crab 3433  Vcvv 3475   class class class wbr 5149   E cep 5580   Se wse 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-se 5633

This theorem is referenced by:  omsinds  7876  omsindsOLD  7877  tfr1ALT  8400  tfr2ALT  8401  tfr3ALT  8402  on2recsfn  8666  on2recsov  8667  on2ind  8668  on3ind  8669  oieu  9534  oismo  9535  oiid  9536  cantnfp1lem3  9675  r0weon  10007  hsmexlem1  10421