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Mirrors > Home > MPE Home > Th. List > exse | Structured version Visualization version GIF version |
Description: Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.) |
Ref | Expression |
---|---|
exse | ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabexg 5226 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
2 | 1 | ralrimivw 3183 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
3 | df-se 5509 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
4 | 2, 3 | sylibr 236 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∀wral 3138 {crab 3142 Vcvv 3494 class class class wbr 5058 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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-se 5509 |
This theorem is referenced by: wemoiso 7668 wemoiso2 7669 oiiso 8995 hartogslem1 9000 oemapwe 9151 cantnffval2 9152 om2uzoi 13317 uzsinds 13349 bpolylem 15396 |
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