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| Mirrors > Home > MPE Home > Th. List > Mathboxes > aiota0ndef | Structured version Visualization version GIF version | ||
| Description: Example for an undefined alternate iota being no set, i.e., ∀𝑦𝑦 ∈ 𝑥 is a wff not satisfied by a (unique) value 𝑥 (there is no set, and therefore certainly no unique set, which contains every set). This is different from iota0ndef 47051, where the iota still is a set (the empty set). (Contributed by AV, 25-Aug-2022.) |
| Ref | Expression |
|---|---|
| aiota0ndef | ⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nalset 5313 | . . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
| 2 | 1 | intnanr 487 | . . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥) |
| 3 | df-eu 2569 | . . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥)) | |
| 4 | 2, 3 | mtbir 323 | . 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥 |
| 5 | df-nel 3047 | . . 3 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V) | |
| 6 | aiotaexb 47101 | . . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V) | |
| 7 | 5, 6 | xchbinxr 335 | . 2 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥) |
| 8 | 4, 7 | mpbir 231 | 1 ⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2538 ∃!weu 2568 ∉ wnel 3046 Vcvv 3480 ℩'caiota 47095 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-int 4947 df-aiota 47097 |
| This theorem is referenced by: (None) |
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