Theorem zfaus 1480

Description: Separation Scheme, which is an axiom scheme of Zermelo's original theory. Scheme Sep of [BellMachover] p. 463. In some textbooks this is given as a separate axiom; here we show it is redundant if we assume ax-rep 1075. The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with x ∈ A) so that it asserts the existence of a collection only if it is smaller than some other collection A that already exists. This prevents Russell's paradox ru 1437. In some texts this scheme is called "Aussonderung" or the Subset Axiom. In typical applications the variable x is free in the wff φ.

Hypothesis

Ref	Expression
zfaus.1	⊢ A ∈ V

Assertion

Ref	Expression
zfaus	⊢ ∃y∀x(x ∈ y ↔ (x ∈ A ∧ φ))

Distinct variable group(s):   x,y,A   φ,y

Proof of Theorem zfaus

Step	Hyp	Ref	Expression
1		zfaus.1	. . 3 ⊢ A ∈ V
2		a9e 809	. . . . 5 ⊢ ∃y y = z
3		eqt 814	. . . . . . . . 9 ⊢ (y = z → (z = x → y = x))
4		eqcom 811	. . . . . . . . 9 ⊢ (y = x → x = y)
5	3, 4	syl6 23	. . . . . . . 8 ⊢ (y = z → (z = x → x = y))
6	5	adantrd 308	. . . . . . 7 ⊢ (y = z → ((z = x ∧ φ) → x = y))
7	6	19.21aiv 943	. . . . . 6 ⊢ (y = z → ∀x((z = x ∧ φ) → x = y))
8	7	19.22i 723	. . . . 5 ⊢ (∃y y = z → ∃y∀x((z = x ∧ φ) → x = y))
9	2, 8	ax-mp 6	. . . 4 ⊢ ∃y∀x((z = x ∧ φ) → x = y)
10	9	a1i 7	. . 3 ⊢ (z ∈ A → ∃y∀x((z = x ∧ φ) → x = y))
11	1, 10	zfrep3cl 1478	. 2 ⊢ ∃y∀x(x ∈ y ↔ ∃z(z ∈ A ∧ (z = x ∧ φ)))
12		an12 370	. . . . . . 7 ⊢ ((z = x ∧ (z ∈ A ∧ φ)) ↔ (z ∈ A ∧ (z = x ∧ φ)))
13	12	biex 733	. . . . . 6 ⊢ (∃z(z = x ∧ (z ∈ A ∧ φ)) ↔ ∃z(z ∈ A ∧ (z = x ∧ φ)))
14		visset 1350	. . . . . . 7 ⊢ x ∈ V
15		eleq1 1149	. . . . . . . 8 ⊢ (z = x → (z ∈ A ↔ x ∈ A))
16	15	anbi1d 469	. . . . . . 7 ⊢ (z = x → ((z ∈ A ∧ φ) ↔ (x ∈ A ∧ φ)))
17	14, 16	ceqsexv 1371	. . . . . 6 ⊢ (∃z(z = x ∧ (z ∈ A ∧ φ)) ↔ (x ∈ A ∧ φ))
18	13, 17	bitr3 153	. . . . 5 ⊢ (∃z(z ∈ A ∧ (z = x ∧ φ)) ↔ (x ∈ A ∧ φ))
19	18	bibi2i 460	. . . 4 ⊢ ((x ∈ y ↔ ∃z(z ∈ A ∧ (z = x ∧ φ))) ↔ (x ∈ y ↔ (x ∈ A ∧ φ)))
20	19	bial 695	. . 3 ⊢ (∀x(x ∈ y ↔ ∃z(z ∈ A ∧ (z = x ∧ φ))) ↔ ∀x(x ∈ y ↔ (x ∈ A ∧ φ)))
21	20	biex 733	. 2 ⊢ (∃y∀x(x ∈ y ↔ ∃z(z ∈ A ∧ (z = x ∧ φ))) ↔ ∃y∀x(x ∈ y ↔ (x ∈ A ∧ φ)))
22	11, 21	mpbi 164	1 ⊢ ∃y∀x(x ∈ y ↔ (x ∈ A ∧ φ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  bm1.3ii 1481  nalset 1482  inex1 1697  zfnul 1746

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-12 802  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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