Theorem aiota0def 47690

Description: Example for a defined alternate 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). This corresponds to iota0def 47632. (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
aiota0def	⊢ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅

Distinct variable group:   𝑥,𝑦

Proof of Theorem aiota0def

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5257	. 2 ⊢ ∅ ∈ V
2		al0ssb 5258	. . 3 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
3	2	ax-gen 1815	. 2 ⊢ ∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
4		eqeq2 2774	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑥 = 𝑧 ↔ 𝑥 = ∅))
5	4	bibi2d 344	. . . . 5 ⊢ (𝑧 = ∅ → ((∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) ↔ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)))
6	5	albidv 1940	. . . 4 ⊢ (𝑧 = ∅ → (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)))
7		eqeq2 2774	. . . 4 ⊢ (𝑧 = ∅ → ((℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = 𝑧 ↔ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅))
8	6, 7	imbi12d 346	. . 3 ⊢ (𝑧 = ∅ → ((∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = 𝑧) ↔ (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅)))
9		aiotaval 47689	. . 3 ⊢ (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = 𝑧)
10	8, 9	vtoclg 3522	. 2 ⊢ (∅ ∈ V → (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅))
11	1, 3, 10	mp2 9	1 ⊢ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  ℩'caiota 47677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-nul 5256

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-sn 4583  df-pr 4585  df-uni 4866  df-int 4906  df-iota 6477  df-aiota 47679

This theorem is referenced by: (None)