Theorem ac2 3567

Description: Axiom of Choice using abbreviations. This cute and very short version does not make use of any defined objects such as the empty set or a function value. However, it is hard to explain intuitively. If you want to figure it out, the rewritten equivalent ac3 3568 is easier to understand. Note: aceq0 3553 shows the logical equivalence to ax-ac 1080.

Assertion

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

Distinct variable group(s):   x,y,z,w,v,u

Proof of Theorem ac2

Step	Hyp	Ref	Expression
1		ax-ac 1080	. 2 ⊢ ∃y∀z∀w((z ∈ w ∧ w ∈ x) → ∃v∀u(∃t((u ∈ w ∧ w ∈ t) ∧ (u ∈ t ∧ t ∈ y)) ↔ u = v))
2		aceq0 3553	. 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 165	1 ⊢ ∃y∀z ∈ x ∀w ∈ z ∃!v ∈ z ∃u ∈ y (z ∈ u ∧ v ∈ u)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803  ∀wral 1201  ∃wrex 1202  ∃!wreu 1203

This theorem is referenced by:  ac3 3568  ac7 3569

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-17 925  ax-ext 1074  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207

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