Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aiota0def Structured version   Visualization version   GIF version

Theorem aiota0def 47108
Description: Example for a defined alternate iota being the empty set, i.e., 𝑦𝑥𝑦 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). This corresponds to iota0def 47050. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
aiota0def (℩'𝑥𝑦 𝑥𝑦) = ∅
Distinct variable group:   𝑥,𝑦

Proof of Theorem aiota0def
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 0ex 5307 . 2 ∅ ∈ V
2 al0ssb 5308 . . 3 (∀𝑦 𝑥𝑦𝑥 = ∅)
32ax-gen 1795 . 2 𝑥(∀𝑦 𝑥𝑦𝑥 = ∅)
4 eqeq2 2749 . . . . . 6 (𝑧 = ∅ → (𝑥 = 𝑧𝑥 = ∅))
54bibi2d 342 . . . . 5 (𝑧 = ∅ → ((∀𝑦 𝑥𝑦𝑥 = 𝑧) ↔ (∀𝑦 𝑥𝑦𝑥 = ∅)))
65albidv 1920 . . . 4 (𝑧 = ∅ → (∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) ↔ ∀𝑥(∀𝑦 𝑥𝑦𝑥 = ∅)))
7 eqeq2 2749 . . . 4 (𝑧 = ∅ → ((℩'𝑥𝑦 𝑥𝑦) = 𝑧 ↔ (℩'𝑥𝑦 𝑥𝑦) = ∅))
86, 7imbi12d 344 . . 3 (𝑧 = ∅ → ((∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) → (℩'𝑥𝑦 𝑥𝑦) = 𝑧) ↔ (∀𝑥(∀𝑦 𝑥𝑦𝑥 = ∅) → (℩'𝑥𝑦 𝑥𝑦) = ∅)))
9 aiotaval 47107 . . 3 (∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) → (℩'𝑥𝑦 𝑥𝑦) = 𝑧)
108, 9vtoclg 3554 . 2 (∅ ∈ V → (∀𝑥(∀𝑦 𝑥𝑦𝑥 = ∅) → (℩'𝑥𝑦 𝑥𝑦) = ∅))
111, 3, 10mp2 9 1 (℩'𝑥𝑦 𝑥𝑦) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  c0 4333  ℩'caiota 47095
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-nul 5306
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-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-int 4947  df-iota 6514  df-aiota 47097
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator