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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabidim2 | Structured version Visualization version GIF version |
Description: Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rabidim2 | ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabid 3455 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | simprbi 496 | 1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 {crab 3433 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 |
This theorem is referenced by: infnsuprnmpt 45195 preimagelt 46655 preimalegt 46656 pimrecltpos 46664 pimiooltgt 46666 pimrecltneg 46680 smfaddlem1 46719 smflimlem2 46728 smfrec 46745 smfmullem4 46750 smfdiv 46753 smfsupxr 46772 smfinflem 46773 smflimsuplem7 46782 smflimsuplem8 46783 fsupdm 46798 finfdm 46802 |
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