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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 5233 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) | |
2 | euabsn2 4653 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
3 | df-aiota 43162 | . . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
4 | 3 | eleq1i 2900 | . 2 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
5 | 1, 2, 4 | 3bitr4i 304 | 1 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∃!weu 2646 {cab 2796 Vcvv 3492 {csn 4557 ∩ cint 4867 ℩'caiota 43160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-int 4868 df-aiota 43162 |
This theorem is referenced by: aiotavb 43167 iotan0aiotaex 43168 aiotaexaiotaiota 43169 aiota0ndef 43172 |
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