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 45318
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 45262. (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 5264 . 2 ∅ ∈ V
2 al0ssb 5265 . . 3 (∀𝑦 𝑥𝑦𝑥 = ∅)
32ax-gen 1797 . 2 𝑥(∀𝑦 𝑥𝑦𝑥 = ∅)
4 eqeq2 2748 . . . . . 6 (𝑧 = ∅ → (𝑥 = 𝑧𝑥 = ∅))
54bibi2d 342 . . . . 5 (𝑧 = ∅ → ((∀𝑦 𝑥𝑦𝑥 = 𝑧) ↔ (∀𝑦 𝑥𝑦𝑥 = ∅)))
65albidv 1923 . . . 4 (𝑧 = ∅ → (∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) ↔ ∀𝑥(∀𝑦 𝑥𝑦𝑥 = ∅)))
7 eqeq2 2748 . . . 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 45317 . . 3 (∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) → (℩'𝑥𝑦 𝑥𝑦) = 𝑧)
108, 9vtoclg 3525 . 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 205  wal 1539   = wceq 1541  wcel 2106  Vcvv 3445  wss 3910  c0 4282  ℩'caiota 45305
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-nul 5263
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-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-sn 4587  df-pr 4589  df-uni 4866  df-int 4908  df-iota 6448  df-aiota 45307
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator