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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 5219 | . . . 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 | iotanul 6335 | . 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 398 ∀wal 1535 = wceq 1537 ∃wex 1780 ∃*wmo 2620 ∃!weu 2653 ∅c0 4293 ℩cio 6314 |
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 2795 ax-sep 5205 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-uni 4841 df-iota 6316 |
This theorem is referenced by: (None) |
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