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Mirrors > Home > MPE Home > Th. List > iotacl | Structured version Visualization version GIF version |
Description: Membership law for
descriptions.
This can be useful for expanding an unbounded iota-based definition (see df-iota 6313). If you have a bounded iota-based definition, riotacl2 7129 may be useful. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Ref | Expression |
---|---|
iotacl | ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota4 6335 | . 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) | |
2 | df-sbc 3772 | . 2 ⊢ ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | sylib 220 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∃!weu 2649 {cab 2799 [wsbc 3771 ℩cio 6311 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3772 df-un 3940 df-in 3942 df-ss 3951 df-sn 4567 df-pr 4569 df-uni 4838 df-iota 6313 |
This theorem is referenced by: riotacl2 7129 opiota 7756 eroprf 8394 iunfictbso 9539 isf32lem9 9782 psgnvali 18635 fourierdlem36 42427 |
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