Theorem epse 5532

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 5463	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
2	1	bicomi 226	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥)
3	2	abbi2i 2953	. . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥}
4		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltrri 2910	. . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V
6		rabssab 4059	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥}
7	5, 6	ssexi 5218	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
8	7	rgenw 3150	. 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
9		df-se 5509	. 2 ⊢ ( E Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V)
10	8, 9	mpbir 233	1 ⊢ E Se 𝐴

Syntax hints:   ∈ wcel 2110  {cab 2799  ∀wral 3138  {crab 3142  Vcvv 3494   class class class wbr 5058   E cep 5458   Se wse 5506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-eprel 5459  df-se 5509

This theorem is referenced by:  omsinds  7594  tfr1ALT  8030  tfr2ALT  8031  tfr3ALT  8032  oieu  8997  oismo  8998  oiid  8999  cantnfp1lem3  9137  r0weon  9432  hsmexlem1  9842