Theorem aiota0def 47284

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 47226. (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 5250	. 2 ⊢ ∅ ∈ V
2		al0ssb 5251	. . 3 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
3	2	ax-gen 1796	. 2 ⊢ ∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
4		eqeq2 2746	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑥 = 𝑧 ↔ 𝑥 = ∅))
5	4	bibi2d 342	. . . . 5 ⊢ (𝑧 = ∅ → ((∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) ↔ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)))
6	5	albidv 1921	. . . 4 ⊢ (𝑧 = ∅ → (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)))
7		eqeq2 2746	. . . 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 47283	. . 3 ⊢ (∀𝑥(∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = 𝑧) → (℩'𝑥∀𝑦 𝑥 ⊆ 𝑦) = 𝑧)
10	8, 9	vtoclg 3509	. 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 1539   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ⊆ wss 3899  ∅c0 4283  ℩'caiota 47271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-sn 4579  df-pr 4581  df-uni 4862  df-int 4901  df-iota 6446  df-aiota 47273

This theorem is referenced by: (None)