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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 5234 | . . 3 ⊢ (¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V) | |
2 | df-aiota 43360 | . . . . 5 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
3 | 2 | eleq1i 2902 | . . . 4 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
4 | 3 | notbii 322 | . . 3 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
5 | 2 | eqeq1i 2825 | . . 3 ⊢ ((℩'𝑥𝜑) = V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V) |
6 | 1, 4, 5 | 3bitr4i 305 | . 2 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V) |
7 | aiotaexb 43364 | . 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) | |
8 | 6, 7 | xchnxbir 335 | 1 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∃!weu 2652 {cab 2798 Vcvv 3491 {csn 4560 ∩ cint 4869 ℩'caiota 43358 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rab 3146 df-v 3493 df-dif 3932 df-in 3936 df-ss 3945 df-nul 4285 df-sn 4561 df-int 4870 df-aiota 43360 |
This theorem is referenced by: dfafv2 43406 |
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