Theorem epse 5605

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 5526	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
2	1	bicomi 224	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥)
3	2	eqabi 2863	. . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥}
4		vex 3442	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltrri 2825	. . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V
6		rabssab 4038	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥}
7	5, 6	ssexi 5264	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
8	7	rgenw 3048	. 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
9		df-se 5577	. 2 ⊢ ( E Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V)
10	8, 9	mpbir 231	1 ⊢ E Se 𝐴

Syntax hints:   ∈ wcel 2109  {cab 2707  ∀wral 3044  {crab 3396  Vcvv 3438   class class class wbr 5095   E cep 5522   Se wse 5574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-eprel 5523  df-se 5577

This theorem is referenced by:  omsinds  7827  tfr1ALT  8329  tfr2ALT  8330  tfr3ALT  8331  on2recsfn  8592  on2recsov  8593  on2ind  8594  on3ind  8595  oieu  9450  oismo  9451  oiid  9452  cantnfp1lem3  9595  r0weon  9925  hsmexlem1  10339  onsse  28194  trfr  44936