Theorem iota0def 47034

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 5282	. 2 ⊢ ∅ ∈ V
2		al0ssb 5283	. . . 4 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
3	2	a1i 11	. . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅))
4	3	iota5 6519	. 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅)
5	1, 1, 4	mp2an 692	1 ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  ℩cio 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-sn 4607  df-pr 4609  df-uni 4889  df-iota 6489

This theorem is referenced by: (None)