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| Mirrors > Home > MPE Home > Th. List > Mathboxes > aiotaexb | Structured version Visualization version GIF version | ||
| Description: The alternate iota over a wff 𝜑 is a set iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.) |
| Ref | Expression |
|---|---|
| aiotaexb | ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | intexab 5346 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) | |
| 2 | euabsn2 4725 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
| 3 | df-aiota 47097 | . . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
| 4 | 3 | eleq1i 2832 | . 2 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
| 5 | 1, 2, 4 | 3bitr4i 303 | 1 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃!weu 2568 {cab 2714 Vcvv 3480 {csn 4626 ∩ cint 4946 ℩'caiota 47095 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-int 4947 df-aiota 47097 |
| This theorem is referenced by: aiotavb 47102 iotan0aiotaex 47105 aiotaexaiotaiota 47106 aiota0ndef 47109 |
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