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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 46047, 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 5312 | . . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | 1 | intnanr 486 | . . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥) |
3 | df-eu 2561 | . . 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 3045 | . . 3 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V) | |
6 | aiotaexb 46095 | . . 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 394 ∀wal 1537 ∃wex 1779 ∈ wcel 2104 ∃*wmo 2530 ∃!weu 2560 ∉ wnel 3044 Vcvv 3472 ℩'caiota 46089 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-int 4950 df-aiota 46091 |
This theorem is referenced by: (None) |
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