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Theorem iota0def 46988
Description: Example for a defined 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). (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
iota0def (℩𝑥𝑦 𝑥𝑦) = ∅
Distinct variable group:   𝑥,𝑦

Proof of Theorem iota0def
StepHypRef Expression
1 0ex 5313 . 2 ∅ ∈ V
2 al0ssb 5314 . . . 4 (∀𝑦 𝑥𝑦𝑥 = ∅)
32a1i 11 . . 3 ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥𝑦𝑥 = ∅))
43iota5 6546 . 2 ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥𝑦 𝑥𝑦) = ∅)
51, 1, 4mp2an 692 1 (℩𝑥𝑦 𝑥𝑦) = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  c0 4339  cio 6514
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-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-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by: (None)
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