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Theorem axsep 4920
Description: Separation Scheme, which is an axiom scheme of Zermelo's original theory. Scheme Sep of [BellMachover] p. 463. As we show here, it is redundant if we assume Replacement in the form of ax-rep 4911. Some textbooks present Separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger Replacement. The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with 𝑥𝑧) so that it asserts the existence of a collection only if it is smaller than some other collection 𝑧 that already exists. This prevents Russell's paradox ru 3563. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

The variable 𝑥 can appear free in the wff 𝜑, which in textbooks is often written 𝜑(𝑥). To specify this in the Metamath language, we omit the distinct variable requirement ($d) that 𝑥 not appear in 𝜑.

For a version using a class variable, see zfauscl 4923, which requires the Axiom of Extensionality as well as Separation for its derivation.

If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 4977 shows (contradicting zfauscl 4923). However, as axsep2 4922 shows, we can eliminate the restriction that 𝑧 not occur in 𝜑.

Note: the distinct variable restriction that 𝑧 not occur in 𝜑 is actually redundant in this particular proof, but we keep it since its purpose is to demonstrate the derivation of the exact ax-sep 4921 from ax-rep 4911.

This theorem should not be referenced by any proof. Instead, use ax-sep 4921 below so that the uses of the Axiom of Separation can be more easily identified. (Contributed by NM, 11-Sep-2006.) (New usage is discouraged.)

Assertion
Ref Expression
axsep 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem axsep
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1980 . . . 4 𝑦(𝑤 = 𝑥𝜑)
21axrep5 4916 . . 3 (∀𝑤(𝑤𝑧 → ∃𝑦𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦)) → ∃𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))))
3 equtr 2091 . . . . . . . 8 (𝑦 = 𝑤 → (𝑤 = 𝑥𝑦 = 𝑥))
4 equcomi 2087 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
53, 4syl6 35 . . . . . . 7 (𝑦 = 𝑤 → (𝑤 = 𝑥𝑥 = 𝑦))
65adantrd 485 . . . . . 6 (𝑦 = 𝑤 → ((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
76alrimiv 1992 . . . . 5 (𝑦 = 𝑤 → ∀𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
87a1d 25 . . . 4 (𝑦 = 𝑤 → (𝑤𝑧 → ∀𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦)))
98spimev 2392 . . 3 (𝑤𝑧 → ∃𝑦𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
102, 9mpg 1861 . 2 𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
11 an12 873 . . . . . . 7 ((𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ (𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
1211exbii 1911 . . . . . 6 (∃𝑤(𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
13 elequ1 2134 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
1413anbi1d 743 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤𝑧𝜑) ↔ (𝑥𝑧𝜑)))
1514equsexvw 2075 . . . . . 6 (∃𝑤(𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ (𝑥𝑧𝜑))
1612, 15bitr3i 266 . . . . 5 (∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)) ↔ (𝑥𝑧𝜑))
1716bibi2i 326 . . . 4 ((𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
1817albii 1884 . . 3 (∀𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)))
1918exbii 1911 . 2 (∃𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)))
2010, 19mpbi 220 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1618  wex 1841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-rep 4911
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847
This theorem is referenced by: (None)
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