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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 5337 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
2 | 1 | ralrimivw 3139 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
3 | df-se 5637 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
4 | 2, 3 | sylibr 233 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∀wral 3050 {crab 3418 Vcvv 3461 class class class wbr 5152 Se wse 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5303 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rab 3419 df-v 3463 df-in 3953 df-ss 3963 df-se 5637 |
This theorem is referenced by: wemoiso 7986 wemoiso2 7987 oiiso 9576 hartogslem1 9581 oemapwe 9733 cantnffval2 9734 om2uzoi 13970 uzsinds 14002 bpolylem 16045 om2noseqoi 28269 |
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