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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 5863 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | |
2 | iotanul 5866 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
3 | ifval 4127 | . 2 ⊢ ((℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) ↔ ((∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅))) | |
4 | 1, 2, 3 | mpbir2an 955 | 1 ⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∃!weu 2470 {cab 2608 ∅c0 3915 ifcif 4086 ∪ cuni 4436 ℩cio 5849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 |
This theorem is referenced by: (None) |
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