![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7383 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
2 | riotaund 7405 | . . . . . 6 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅) | |
3 | 2 | eleq1d 2819 | . . . . 5 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ∅ ∈ 𝐴)) |
4 | 3 | notbid 318 | . . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ¬ ∅ ∈ 𝐴)) |
5 | 4 | biimprcd 249 | . . 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 225 | 1 ⊢ (¬ ∅ ∈ 𝐴 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2107 ∃!wreu 3375 ∅c0 4323 ℩crio 7364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-sn 4630 df-pr 4632 df-uni 4910 df-iota 6496 df-riota 7365 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |