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Theorem axsep 4702
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 4693. 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 3400. 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 4705, 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 4761 shows (contradicting zfauscl 4705). However, as axsep2 4704 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 4703 from ax-rep 4693.

This theorem should not be referenced by any proof. Instead, use ax-sep 4703 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 1829 . . . 4 𝑦(𝑤 = 𝑥𝜑)
21axrep5 4698 . . 3 (∀𝑤(𝑤𝑧 → ∃𝑦𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦)) → ∃𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))))
3 equtr 1934 . . . . . . . 8 (𝑦 = 𝑤 → (𝑤 = 𝑥𝑦 = 𝑥))
4 equcomi 1930 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
53, 4syl6 34 . . . . . . 7 (𝑦 = 𝑤 → (𝑤 = 𝑥𝑥 = 𝑦))
65adantrd 482 . . . . . 6 (𝑦 = 𝑤 → ((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
76alrimiv 1841 . . . . 5 (𝑦 = 𝑤 → ∀𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
87a1d 25 . . . 4 (𝑦 = 𝑤 → (𝑤𝑧 → ∀𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦)))
98spimev 2246 . . 3 (𝑤𝑧 → ∃𝑦𝑥((𝑤 = 𝑥𝜑) → 𝑥 = 𝑦))
102, 9mpg 1714 . 2 𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
11 an12 833 . . . . . . 7 ((𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ (𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
1211exbii 1763 . . . . . 6 (∃𝑤(𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)))
13 elequ1 1983 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
1413anbi1d 736 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤𝑧𝜑) ↔ (𝑥𝑧𝜑)))
1514equsexvw 1918 . . . . . 6 (∃𝑤(𝑤 = 𝑥 ∧ (𝑤𝑧𝜑)) ↔ (𝑥𝑧𝜑))
1612, 15bitr3i 264 . . . . 5 (∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑)) ↔ (𝑥𝑧𝜑))
1716bibi2i 325 . . . 4 ((𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
1817albii 1736 . . 3 (∀𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)))
1918exbii 1763 . 2 (∃𝑦𝑥(𝑥𝑦 ↔ ∃𝑤(𝑤𝑧 ∧ (𝑤 = 𝑥𝜑))) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)))
2010, 19mpbi 218 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-rep 4693
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by: (None)
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