Theorem iota0def 47816

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	⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅

Distinct variable group:   𝑥,𝑦

Proof of Theorem iota0def

Step	Hyp	Ref	Expression
1		0ex 5275	. 2 ⊢ ∅ ∈ V
2		al0ssb 5276	. . . 4 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
3	2	a1i 11	. . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅))
4	3	iota5 6526	. 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅)
5	1, 1, 4	mp2an 705	1 ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ⊆ wss 3908  ∅c0 4289  ℩cio 6497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-sn 4595  df-pr 4597  df-uni 4878  df-iota 6499

This theorem is used by: (None)