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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.) |
Ref | Expression |
---|---|
dfiota4 | ⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotauni 5780 | . . 3 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) | |
2 | iftrue 4042 | . . 3 ⊢ (∃!𝑥𝜑 → if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) = ∪ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | eqtr4d 2647 | . 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅)) |
4 | iotanul 5783 | . . 3 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅) | |
5 | iffalse 4045 | . . 3 ⊢ (¬ ∃!𝑥𝜑 → if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) = ∅) | |
6 | 4, 5 | eqtr4d 2647 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅)) |
7 | 3, 6 | pm2.61i 175 | 1 ⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∃!weu 2458 {cab 2596 ∅c0 3874 ifcif 4036 ∪ cuni 4372 ℩cio 5766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 |
This theorem is referenced by: (None) |
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