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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iota0ndef | Structured version Visualization version GIF version | ||
| Description: Example for an undefined iota being the empty 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). (Contributed by AV, 24-Aug-2022.) |
| Ref | Expression |
|---|---|
| iota0ndef | ⊢ (℩𝑥∀𝑦 𝑦 ∈ 𝑥) = ∅ |
| 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 | iotanul 6539 | . 2 ⊢ (¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥 → (℩𝑥∀𝑦 𝑦 ∈ 𝑥) = ∅) | |
| 6 | 4, 5 | ax-mp 5 | 1 ⊢ (℩𝑥∀𝑦 𝑦 ∈ 𝑥) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∃*wmo 2538 ∃!weu 2568 ∅c0 4333 ℩cio 6512 |
| 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-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-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-sn 4627 df-uni 4908 df-iota 6514 |
| This theorem is referenced by: (None) |
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