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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 3458 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 {crab 3436 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 |
| This theorem is referenced by: frgrwopreglem5 30340 frgrwopreg 30342 rabexgfGS 32518 ssrab2f 45122 infnsuprnmpt 45257 preimagelt 46714 preimalegt 46715 pimrecltpos 46723 pimrecltneg 46739 smfresal 46803 smfpimbor1lem2 46814 smflimmpt 46825 smfsupmpt 46830 smfinfmpt 46834 smflimsuplem7 46841 smflimsuplem8 46842 smflimsupmpt 46844 smfliminfmpt 46847 fsupdm 46857 finfdm 46861 |
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