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Theorem iota0def 46317
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 5300 . 2 ∅ ∈ V
2 al0ssb 5301 . . . 4 (∀𝑦 𝑥𝑦𝑥 = ∅)
32a1i 11 . . 3 ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥𝑦𝑥 = ∅))
43iota5 6520 . 2 ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥𝑦 𝑥𝑦) = ∅)
51, 1, 4mp2an 689 1 (℩𝑥𝑦 𝑥𝑦) = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wal 1531   = wceq 1533  wcel 2098  Vcvv 3468  wss 3943  c0 4317  cio 6487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-uni 4903  df-iota 6489
This theorem is referenced by: (None)
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