Theorem iota0def 45738

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 5307	. 2 ⊢ ∅ ∈ V
2		al0ssb 5308	. . . 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 690	1 ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅

Syntax hints:   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  ℩cio 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495

This theorem is referenced by: (None)