Metamath Proof Explorer


Theorem aceq0

Description: Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac . (Contributed by NM, 5-Apr-2004)

Ref Expression
Assertion aceq0 ( ∃ 𝑦𝑧𝑥𝑤𝑧 ∃! 𝑣𝑧𝑢𝑦 ( 𝑧𝑢𝑣𝑢 ) ↔ ∃ 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) )

Proof

Step Hyp Ref Expression
1 aceq1 ( ∃ 𝑦𝑧𝑥𝑤𝑧 ∃! 𝑣𝑧𝑢𝑦 ( 𝑧𝑢𝑣𝑢 ) ↔ ∃ 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
2 equequ2 ( 𝑣 = 𝑥 → ( 𝑢 = 𝑣𝑢 = 𝑥 ) )
3 2 bibi2d ( 𝑣 = 𝑥 → ( ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
4 elequ2 ( 𝑡 = 𝑥 → ( 𝑤𝑡𝑤𝑥 ) )
5 4 anbi2d ( 𝑡 = 𝑥 → ( ( 𝑢𝑤𝑤𝑡 ) ↔ ( 𝑢𝑤𝑤𝑥 ) ) )
6 elequ2 ( 𝑡 = 𝑥 → ( 𝑢𝑡𝑢𝑥 ) )
7 elequ1 ( 𝑡 = 𝑥 → ( 𝑡𝑦𝑥𝑦 ) )
8 6 7 anbi12d ( 𝑡 = 𝑥 → ( ( 𝑢𝑡𝑡𝑦 ) ↔ ( 𝑢𝑥𝑥𝑦 ) ) )
9 5 8 anbi12d ( 𝑡 = 𝑥 → ( ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ) )
10 9 cbvexvw ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) )
11 10 bibi1i ( ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) )
12 3 11 bitrdi ( 𝑣 = 𝑥 → ( ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
13 12 albidv ( 𝑣 = 𝑥 → ( ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∀ 𝑢 ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
14 elequ1 ( 𝑢 = 𝑧 → ( 𝑢𝑤𝑧𝑤 ) )
15 14 anbi1d ( 𝑢 = 𝑧 → ( ( 𝑢𝑤𝑤𝑥 ) ↔ ( 𝑧𝑤𝑤𝑥 ) ) )
16 elequ1 ( 𝑢 = 𝑧 → ( 𝑢𝑥𝑧𝑥 ) )
17 16 anbi1d ( 𝑢 = 𝑧 → ( ( 𝑢𝑥𝑥𝑦 ) ↔ ( 𝑧𝑥𝑥𝑦 ) ) )
18 15 17 anbi12d ( 𝑢 = 𝑧 → ( ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ) )
19 18 exbidv ( 𝑢 = 𝑧 → ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ) )
20 equequ1 ( 𝑢 = 𝑧 → ( 𝑢 = 𝑥𝑧 = 𝑥 ) )
21 19 20 bibi12d ( 𝑢 = 𝑧 → ( ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
22 21 cbvalvw ( ∀ 𝑢 ( ∃ 𝑥 ( ( 𝑢𝑤𝑤𝑥 ) ∧ ( 𝑢𝑥𝑥𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) )
23 13 22 bitrdi ( 𝑣 = 𝑥 → ( ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
24 23 cbvexvw ( ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) )
25 24 imbi2i ( ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
26 25 2albii ( ∀ 𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ∀ 𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
27 26 exbii ( ∃ 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ∃ 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑥𝑧 ( ∃ 𝑥 ( ( 𝑧𝑤𝑤𝑥 ) ∧ ( 𝑧𝑥𝑥𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
28 1 27 bitr4i ( ∃ 𝑦𝑧𝑥𝑤𝑧 ∃! 𝑣𝑧𝑢𝑦 ( 𝑧𝑢𝑣𝑢 ) ↔ ∃ 𝑦𝑧𝑤 ( ( 𝑧𝑤𝑤𝑥 ) → ∃ 𝑣𝑢 ( ∃ 𝑡 ( ( 𝑢𝑤𝑤𝑡 ) ∧ ( 𝑢𝑡𝑡𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) )