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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 5433 | . . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
2 | 1 | bicomi 227 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥) |
3 | 2 | abbi2i 2929 | . . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥} |
4 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | eqeltrri 2887 | . . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V |
6 | rabssab 4011 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥} | |
7 | 5, 6 | ssexi 5190 | . . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V |
8 | 7 | rgenw 3118 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V |
9 | df-se 5479 | . 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 2111 {cab 2776 ∀wral 3106 {crab 3110 Vcvv 3441 class class class wbr 5030 E cep 5429 Se wse 5476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-se 5479 |
This theorem is referenced by: omsinds 7580 tfr1ALT 8019 tfr2ALT 8020 tfr3ALT 8021 oieu 8987 oismo 8988 oiid 8989 cantnfp1lem3 9127 r0weon 9423 hsmexlem1 9837 |
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