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Mirrors > Home > MPE Home > Th. List > dfiota4 | Structured version Visualization version GIF version |
Description: The ℩ operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.) |
Ref | Expression |
---|---|
dfiota4 | ⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotauni 6299 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | |
2 | iotanul 6302 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
3 | ifval 4466 | . 2 ⊢ ((℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) ↔ ((∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅))) | |
4 | 1, 2, 3 | mpbir2an 710 | 1 ⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∃!weu 2628 {cab 2776 ∅c0 4243 ifcif 4425 ∪ cuni 4800 ℩cio 6281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 |
This theorem is referenced by: (None) |
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