Theorem aiota0def 47125

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 47067. (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 5277	. 2 ⊢ ∅ ∈ V
2		al0ssb 5278	. . 3 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
3	2	ax-gen 1795	. 2 ⊢ ∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
4		eqeq2 2747	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑥 = 𝑧 ↔ 𝑥 = ∅))
5	4	bibi2d 342	. . . . 5 ⊢ (𝑧 = ∅ → ((∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) ↔ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)))
6	5	albidv 1920	. . . 4 ⊢ (𝑧 = ∅ → (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)))
7		eqeq2 2747	. . . 4 ⊢ (𝑧 = ∅ → ((℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = 𝑧 ↔ (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅))
8	6, 7	imbi12d 344	. . 3 ⊢ (𝑧 = ∅ → ((∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = 𝑧) ↔ (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅)))
9		aiotaval 47124	. . 3 ⊢ (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = 𝑧)
10	8, 9	vtoclg 3533	. 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 206  ∀wal 1538   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ⊆ wss 3926  ∅c0 4308  ℩'caiota 47112

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-sn 4602  df-pr 4604  df-uni 4884  df-int 4923  df-iota 6484  df-aiota 47114

This theorem is referenced by: (None)