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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 5264 | . . 3 ⊢ (¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V) | |
2 | df-aiota 44544 | . . . . 5 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
3 | 2 | eleq1i 2829 | . . . 4 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
4 | 3 | notbii 320 | . . 3 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
5 | 2 | eqeq1i 2743 | . . 3 ⊢ ((℩'𝑥𝜑) = V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V) |
6 | 1, 4, 5 | 3bitr4i 303 | . 2 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V) |
7 | aiotaexb 44548 | . 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) | |
8 | 6, 7 | xchnxbir 333 | 1 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ∃!weu 2568 {cab 2715 Vcvv 3431 {csn 4563 ∩ cint 4881 ℩'caiota 44542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rab 3073 df-v 3433 df-dif 3891 df-in 3895 df-ss 3905 df-nul 4259 df-sn 4564 df-int 4882 df-aiota 44544 |
This theorem is referenced by: dfaiota3 44551 dfafv2 44591 |
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