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| Mirrors > Home > MPE Home > Th. List > riotaclb | Structured version Visualization version GIF version | ||
| Description: Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.) |
| Ref | Expression |
|---|---|
| riotaclb | ⊢ (¬ ∅ ∈ 𝐴 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | riotacl 7370 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
| 2 | riotaund 7392 | . . . . . 6 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅) | |
| 3 | 2 | eleq1d 2847 | . . . . 5 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
| 4 | 3 | notbid 320 | . . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ¬ ∅ ∈ 𝐴)) |
| 5 | 4 | biimprcd 252 | . . 3 ⊢ (¬ ∅ ∈ 𝐴 → (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) |
| 6 | 5 | con4d 115 | . 2 ⊢ (¬ ∅ ∈ 𝐴 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → ∃!𝑥 ∈ 𝐴 𝜑)) |
| 7 | 1, 6 | impbid2 228 | 1 ⊢ (¬ ∅ ∈ 𝐴 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∈ wcel 2142 ∃!wreu 3365 ∅c0 4285 ℩crio 7352 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-dif 3907 df-un 3909 df-ss 3921 df-nul 4286 df-sn 4583 df-pr 4585 df-uni 4866 df-iota 6477 df-riota 7353 |
| This theorem is referenced by: (None) |
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