Theorem ac2 4729

Description: Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single conjunction. (If you want to figure it out, the rewritten equivalent ac3 4730 is easier to understand.) Note: aceq0 4713 shows the logical equivalence to ax-ac 4727.

Assertion

Ref	Expression
ac2	⊢ ∃y∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ⋀ v ∈ u)

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

Proof of Theorem ac2

Step	Hyp	Ref	Expression
1		ax-ac 4727	. 2 ⊢ ∃y∀z∀w((z ∈ w ⋀ w ∈ x) → ∃v∀u(∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v))
2		aceq0 4713	. 2 ⊢ (∃y∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ⋀ v ∈ u) ↔ ∃y∀z∀w((z ∈ w ⋀ w ∈ x) → ∃v∀u(∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v)))
3	1, 2	mpbir 190	1 ⊢ ∃y∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ⋀ v ∈ u)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 953   = wceq 955   ∈ wcel 957  ∃wex 979  ∀wral 1643  ∃wrex 1644  ∃!wreu 1645

This theorem is referenced by:  ac3 4730  ac7 4731

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-11o 1217  ax-ext 1458  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-cleq 1468  df-clel 1471  df-ral 1647  df-rex 1648  df-reu 1649

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