Theorem aceq2 10160

Description: Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
aceq2	⊢ (∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Proof of Theorem aceq2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 3061	. . . . 5 ⊢ (∀𝑡 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∀𝑡(𝑡 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
2		19.23v 1941	. . . . 5 ⊢ (∀𝑡(𝑡 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ↔ (∃𝑡 𝑡 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
3	1, 2	bitri 275	. . . 4 ⊢ (∀𝑡 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ (∃𝑡 𝑡 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
4		biidd 262	. . . . 5 ⊢ (𝑤 = 𝑡 → (∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
5	4	cbvralvw 3236	. . . 4 ⊢ (∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∀𝑡 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
6		n0 4352	. . . . 5 ⊢ (𝑧 ≠ ∅ ↔ ∃𝑡 𝑡 ∈ 𝑧)
7		elequ2 2122	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ 𝑢))
8		elequ2 2122	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ 𝑢))
9	7, 8	anbi12d 632	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → ((𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑢 ∧ 𝑤 ∈ 𝑢)))
10	9	cbvrexvw 3237	. . . . . . 7 ⊢ (∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑤 ∈ 𝑢))
11	10	reubii 3388	. . . . . 6 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃!𝑤 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑤 ∈ 𝑢))
12		elequ1 2114	. . . . . . . . 9 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑢 ↔ 𝑣 ∈ 𝑢))
13	12	anbi2d 630	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → ((𝑧 ∈ 𝑢 ∧ 𝑤 ∈ 𝑢) ↔ (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
14	13	rexbidv 3178	. . . . . . 7 ⊢ (𝑤 = 𝑣 → (∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑤 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
15	14	cbvreuvw 3403	. . . . . 6 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑤 ∈ 𝑢) ↔ ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
16	11, 15	bitri 275	. . . . 5 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
17	6, 16	imbi12i 350	. . . 4 ⊢ ((𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ (∃𝑡 𝑡 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
18	3, 5, 17	3bitr4i 303	. . 3 ⊢ (∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
19	18	ralbii 3092	. 2 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
20	19	exbii 1847	1 ⊢ (∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537  ∃wex 1778   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  ∃!wreu 3377  ∅c0 4332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-dif 3953  df-nul 4333

This theorem is referenced by:  dfac7  10174  ac3  10503