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Theorem aiota0def 47046
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 46988. (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 5313 . 2 ∅ ∈ V
2 al0ssb 5314 . . 3 (∀𝑦 𝑥𝑦𝑥 = ∅)
32ax-gen 1792 . 2 𝑥(∀𝑦 𝑥𝑦𝑥 = ∅)
4 eqeq2 2747 . . . . . 6 (𝑧 = ∅ → (𝑥 = 𝑧𝑥 = ∅))
54bibi2d 342 . . . . 5 (𝑧 = ∅ → ((∀𝑦 𝑥𝑦𝑥 = 𝑧) ↔ (∀𝑦 𝑥𝑦𝑥 = ∅)))
65albidv 1918 . . . 4 (𝑧 = ∅ → (∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) ↔ ∀𝑥(∀𝑦 𝑥𝑦𝑥 = ∅)))
7 eqeq2 2747 . . . 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 47045 . . 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 1535   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  c0 4339  ℩'caiota 47033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-int 4952  df-iota 6516  df-aiota 47035
This theorem is referenced by: (None)
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