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Theorem iota0def 47055
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 5306 . 2 ∅ ∈ V
2 al0ssb 5307 . . . 4 (∀𝑦 𝑥𝑦𝑥 = ∅)
32a1i 11 . . 3 ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥𝑦𝑥 = ∅))
43iota5 6543 . 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 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)
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