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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 43323, 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 5217 | . . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | 1 | intnanr 490 | . . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥) |
3 | df-eu 2654 | . . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥)) | |
4 | 2, 3 | mtbir 325 | . 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥 |
5 | df-nel 3124 | . . 3 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V) | |
6 | aiotaexb 43338 | . . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∈ V) | |
7 | 5, 6 | xchbinxr 337 | . 2 ⊢ ((℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V ↔ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥) |
8 | 4, 7 | mpbir 233 | 1 ⊢ (℩'𝑥∀𝑦 𝑦 ∈ 𝑥) ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∀wal 1535 ∃wex 1780 ∈ wcel 2114 ∃*wmo 2620 ∃!weu 2653 ∉ wnel 3123 Vcvv 3494 ℩'caiota 43332 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 df-sn 4568 df-int 4877 df-aiota 43334 |
This theorem is referenced by: (None) |
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