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Mirrors > Home > MPE Home > Th. List > rabidim1 | Structured version Visualization version GIF version |
Description: Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rabidim1 | ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabid 3116 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 {crab 2916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-tru 1486 df-ex 1705 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-rab 2921 |
This theorem is referenced by: frgrwopreglem5 27185 frgrwopreg 27187 ssrab2f 39300 infnsuprnmpt 39465 pimrecltpos 40919 pimrecltneg 40933 smfresal 40995 smfpimbor1lem2 41006 smflimmpt 41016 smfsupmpt 41021 smfinfmpt 41025 smflimsuplem7 41032 smflimsuplem8 41033 smflimsupmpt 41035 smfliminfmpt 41038 |
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