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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 5205 | . . 3 ⊢ (¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V) | |
2 | df-aiota 43642 | . . . . 5 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
3 | 2 | eleq1i 2880 | . . . 4 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
4 | 3 | notbii 323 | . . 3 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ ¬ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V) |
5 | 2 | eqeq1i 2803 | . . 3 ⊢ ((℩'𝑥𝜑) = V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = V) |
6 | 1, 4, 5 | 3bitr4i 306 | . 2 ⊢ (¬ (℩'𝑥𝜑) ∈ V ↔ (℩'𝑥𝜑) = V) |
7 | aiotaexb 43646 | . 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) | |
8 | 6, 7 | xchnxbir 336 | 1 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃!weu 2628 {cab 2776 Vcvv 3441 {csn 4525 ∩ cint 4838 ℩'caiota 43640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-int 4839 df-aiota 43642 |
This theorem is referenced by: dfafv2 43688 |
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