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 5499 | . . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
2 | 1 | bicomi 223 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥) |
3 | 2 | abbi2i 2880 | . . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥} |
4 | vex 3437 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | eqeltrri 2837 | . . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V |
6 | rabssab 4019 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥} | |
7 | 5, 6 | ssexi 5247 | . . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V |
8 | 7 | rgenw 3077 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V |
9 | df-se 5546 | . 2 ⊢ ( E Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V) | |
10 | 8, 9 | mpbir 230 | 1 ⊢ E Se 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 {cab 2716 ∀wral 3065 {crab 3069 Vcvv 3433 class class class wbr 5075 E cep 5495 Se wse 5543 |
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 2710 ax-sep 5224 ax-nul 5231 ax-pr 5353 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2069 df-clab 2717 df-cleq 2731 df-clel 2817 df-ne 2945 df-ral 3070 df-rab 3074 df-v 3435 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-br 5076 df-opab 5138 df-eprel 5496 df-se 5546 |
This theorem is referenced by: omsinds 7742 omsindsOLD 7743 tfr1ALT 8240 tfr2ALT 8241 tfr3ALT 8242 oieu 9307 oismo 9308 oiid 9309 cantnfp1lem3 9447 r0weon 9777 hsmexlem1 10191 on2recsfn 33835 on2recsov 33836 on2ind 33837 on3ind 33838 |
Copyright terms: Public domain | W3C validator |