|   | Mathbox for Alexander van der Vekens | < Previous  
      Next > Nearby theorems | |
| 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 5306 | . 2 ⊢ ∅ ∈ V | |
| 2 | al0ssb 5307 | . . . 4 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅) | |
| 3 | 2 | a1i 11 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)) | 
| 4 | 3 | iota5 6543 | . 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅) | 
| 5 | 1, 1, 4 | mp2an 692 | 1 ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 ∅c0 4332 ℩cio 6511 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-sn 4626 df-pr 4628 df-uni 4907 df-iota 6513 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |