Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > epse | Structured version Visualization version GIF version |
Description: The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon 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 4952 | . . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
2 | 1 | bicomi 213 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥) |
3 | 2 | abbi2i 2725 | . . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥} |
4 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | eqeltrri 2685 | . . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V |
6 | rabssab 3652 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥} | |
7 | 5, 6 | ssexi 4731 | . . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V |
8 | 7 | rgenw 2908 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V |
9 | df-se 4998 | . 2 ⊢ ( E Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V) | |
10 | 8, 9 | mpbir 220 | 1 ⊢ E Se 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 {cab 2596 ∀wral 2896 {crab 2900 Vcvv 3173 class class class wbr 4583 E cep 4947 Se wse 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-eprel 4949 df-se 4998 |
This theorem is referenced by: omsinds 6976 tfr1ALT 7383 tfr2ALT 7384 tfr3ALT 7385 oieu 8327 oismo 8328 oiid 8329 cantnfp1lem3 8460 r0weon 8718 hsmexlem1 9131 |
Copyright terms: Public domain | W3C validator |