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Mirrors > Home > ILE Home > Th. List > exse | 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 3900 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
2 | 1 | ralrimivw 2393 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
3 | df-se 4070 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
4 | 2, 3 | sylibr 137 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ∀wral 2306 {crab 2310 Vcvv 2557 class class class wbr 3764 Se wse 4066 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rab 2315 df-v 2559 df-in 2924 df-ss 2931 df-se 4070 |
This theorem is referenced by: (None) |
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