Theorem epse 5606

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 5527	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
2	1	bicomi 224	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥)
3	2	eqabi 2872	. . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥}
4		vex 3434	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltrri 2834	. . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V
6		rabssab 4026	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥}
7	5, 6	ssexi 5259	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
8	7	rgenw 3056	. 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
9		df-se 5578	. 2 ⊢ ( E Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V)
10	8, 9	mpbir 231	1 ⊢ E Se 𝐴

Syntax hints:   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3390  Vcvv 3430   class class class wbr 5086   E cep 5523   Se wse 5575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-eprel 5524  df-se 5578

This theorem is referenced by:  omsinds  7831  tfr1ALT  8332  tfr2ALT  8333  tfr3ALT  8334  on2recsfn  8596  on2recsov  8597  on2ind  8598  on3ind  8599  oieu  9447  oismo  9448  oiid  9449  cantnfp1lem3  9592  r0weon  9925  hsmexlem1  10339  onsse  28279  trfr  45407