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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 5268 | . . . 4 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | 1 | intnanr 488 | . . 3 ⊢ ¬ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥) |
3 | df-eu 2567 | . . 3 ⊢ (∃!𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 𝑦 ∈ 𝑥)) | |
4 | 2, 3 | mtbir 322 | . 2 ⊢ ¬ ∃!𝑥∀𝑦 𝑦 ∈ 𝑥 |
5 | iotanul 6471 | . 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 396 ∀wal 1539 = wceq 1541 ∃wex 1781 ∃*wmo 2536 ∃!weu 2566 ∅c0 4280 ℩cio 6443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-v 3445 df-dif 3911 df-in 3915 df-ss 3925 df-nul 4281 df-sn 4585 df-uni 4864 df-iota 6445 |
This theorem is referenced by: (None) |
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