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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 46049, 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 5314 | . . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | 1 | intnanr 487 | . . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥) |
3 | df-eu 2562 | . . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥)) | |
4 | 2, 3 | mtbir 322 | . 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥 |
5 | df-nel 3046 | . . 3 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V) | |
6 | aiotaexb 46097 | . . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V) | |
7 | 5, 6 | xchbinxr 334 | . 2 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥) |
8 | 4, 7 | mpbir 230 | 1 ⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1538 ∃wex 1780 ∈ wcel 2105 ∃*wmo 2531 ∃!weu 2561 ∉ wnel 3045 Vcvv 3473 ℩'caiota 46091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4324 df-sn 4630 df-int 4952 df-aiota 46093 |
This theorem is referenced by: (None) |
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