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Mirrors > Home > MPE Home > Th. List > riotaund | Structured version Visualization version GIF version |
Description: Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.) |
Ref | Expression |
---|---|
riotaund | ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-riota 7382 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | df-reu 3375 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | iotanul 6531 | . . 3 ⊢ (¬ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∅) | |
4 | 2, 3 | sylnbi 329 | . 2 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∅) |
5 | 1, 4 | eqtrid 2780 | 1 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃!weu 2557 ∃!wreu 3372 ∅c0 4326 ℩cio 6503 ℩crio 7381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-reu 3375 df-v 3475 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4327 df-sn 4633 df-uni 4913 df-iota 6505 df-riota 7382 |
This theorem is referenced by: riotassuni 7423 riotaclb 7424 supval2 9486 lubval 18355 glbval 18368 grpinvfval 18942 finxpreclem4 36906 |
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