Theorem aiota0def 46255

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 46199. (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 5297	. 2 ⊢ ∅ ∈ V
2		al0ssb 5298	. . 3 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
3	2	ax-gen 1789	. 2 ⊢ ∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
4		eqeq2 2736	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑥 = 𝑧 ↔ 𝑥 = ∅))
5	4	bibi2d 342	. . . . 5 ⊢ (𝑧 = ∅ → ((∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) ↔ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)))
6	5	albidv 1915	. . . 4 ⊢ (𝑧 = ∅ → (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)))
7		eqeq2 2736	. . . 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 46254	. . 3 ⊢ (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = 𝑧)
10	8, 9	vtoclg 3535	. 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 205  ∀wal 1531   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ⊆ wss 3940  ∅c0 4314  ℩'caiota 46242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-nul 5296

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-sn 4621  df-pr 4623  df-uni 4900  df-int 4941  df-iota 6485  df-aiota 46244

This theorem is referenced by: (None)