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Mirrors > Home > MPE Home > Th. List > Mathboxes > iota0def | Structured version Visualization version GIF version |
Description: Example for a defined iota being the empty set, i.e., ∀𝑦𝑥 ⊆ 𝑦 is a wff satisfied by a unique value 𝑥, namely 𝑥 = ∅ (the empty set is the one and only set which is a subset of every set). (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
iota0def | ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5300 | . 2 ⊢ ∅ ∈ V | |
2 | al0ssb 5301 | . . . 4 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅) | |
3 | 2 | a1i 11 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)) |
4 | 3 | iota5 6520 | . 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅) |
5 | 1, 1, 4 | mp2an 689 | 1 ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 ∅c0 4317 ℩cio 6487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-sn 4624 df-pr 4626 df-uni 4903 df-iota 6489 |
This theorem is referenced by: (None) |
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