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Theorem iota0def 42624
 Description: Example for a defined iota being the empty set, i.e., ∀𝑦𝑥 ⊆ 𝑦 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 5062 . 2 ∅ ∈ V
2 al0ssb 5063 . . . 4 (∀𝑦 𝑥𝑦𝑥 = ∅)
32a1i 11 . . 3 ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥𝑦𝑥 = ∅))
43iota5 6166 . 2 ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥𝑦 𝑥𝑦) = ∅)
51, 1, 4mp2an 679 1 (℩𝑥𝑦 𝑥𝑦) = ∅
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 387  ∀wal 1505   = wceq 1507   ∈ wcel 2048  Vcvv 3409   ⊆ wss 3825  ∅c0 4173  ℩cio 6144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745  ax-nul 5061 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rex 3088  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-sn 4436  df-pr 4438  df-uni 4707  df-iota 6146 This theorem is referenced by: (None)
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