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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aiotavb | Structured version Visualization version GIF version |
Description: The alternate iota over a wff 𝜑 is the universe iff there is no unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.) |
Ref | Expression |
---|---|
aiotavb | ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intnex 5335 | . . 3 ⊢ (¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V) | |
2 | df-aiota 46731 | . . . . 5 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
3 | 2 | eleq1i 2817 | . . . 4 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
4 | 3 | notbii 319 | . . 3 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
5 | 2 | eqeq1i 2731 | . . 3 ⊢ ((℩'𝑥𝜑) = V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V) |
6 | 1, 4, 5 | 3bitr4i 302 | . 2 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V) |
7 | aiotaexb 46735 | . 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) | |
8 | 6, 7 | xchnxbir 332 | 1 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∃!weu 2557 {cab 2703 Vcvv 3462 {csn 4623 ∩ cint 4946 ℩'caiota 46729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-in 3953 df-ss 3963 df-nul 4323 df-sn 4624 df-int 4947 df-aiota 46731 |
This theorem is referenced by: dfaiota3 46738 dfafv2 46778 |
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