Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > epse | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
epse | ⊢ E Se 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epel 5447 | . . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
2 | 1 | bicomi 227 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥) |
3 | 2 | abbi2i 2872 | . . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥} |
4 | vex 3404 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | eqeltrri 2831 | . . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V |
6 | rabssab 3984 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥} | |
7 | 5, 6 | ssexi 5200 | . . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V |
8 | 7 | rgenw 3066 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V |
9 | df-se 5494 | . 2 ⊢ ( E Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V) | |
10 | 8, 9 | mpbir 234 | 1 ⊢ E Se 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 {cab 2717 ∀wral 3054 {crab 3058 Vcvv 3400 class class class wbr 5040 E cep 5443 Se wse 5491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-ral 3059 df-rab 3063 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-eprel 5444 df-se 5494 |
This theorem is referenced by: omsinds 7632 tfr1ALT 8078 tfr2ALT 8079 tfr3ALT 8080 oieu 9089 oismo 9090 oiid 9091 cantnfp1lem3 9229 r0weon 9525 hsmexlem1 9939 on2recsfn 33482 on2recsov 33483 on2ind 33484 on3ind 33485 |
Copyright terms: Public domain | W3C validator |