Definition df-cleq 2615

Description: Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs.

This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce y = z <-> A. x ( x e. y <-> x e. z ), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2606). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated.

See also comments under df-clab 2609, df-clel 2618, and abeq2 2732.

In the form of dfcleq 2616, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

While the three class definitions df-clab 2609, df-cleq 2615, and df-clel 2618 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 15-Sep-1993.)

Hypothesis

Ref	Expression
df-cleq.1	|- ( A. x ( x e. y <-> x e. z ) -> y = z )

Assertion

Ref	Expression
df-cleq	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Distinct variable groups:   x, A   x, B   x, y, z

Allowed substitution hints:   A( y, z)   B( y, z)

Detailed syntax breakdown of Definition df-cleq

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3	1, 2	wceq 1483	. 2 wff A = B
4		vx	. . . . . 6 setvar x
5	4	cv 1482	. . . . 5 class x
6	5, 1	wcel 1990	. . . 4 wff x e. A
7	5, 2	wcel 1990	. . . 4 wff x e. B
8	6, 7	wb 196	. . 3 wff ( x e. A <-> x e. B )
9	8, 4	wal 1481	. 2 wff A. x ( x e. A <-> x e. B )
10	3, 9	wb 196	1 wff ( A = B <-> A. x ( x e. A <-> x e. B ) )

This definition is referenced by:  dfcleq  2616  bj-ax9  32890  bj-ax9-2  32891