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Mirrors > Home > MPE Home > Th. List > dfcleq  Unicode version 
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
We could avoid this complication by introducing a new symbol, say
=_{2},
in place of However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality. See also comments under dfclab 2443, dfclel 2452, and abeq2 2581. In the form of dfcleq 2450, 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 dfclab 2443, dfcleq 2449, and dfclel 2452 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 http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 15Sep1993.) 
Ref  Expression 

dfcleq.1 
Ref  Expression 

dfcleq 
Step  Hyp  Ref  Expression 

1  cA  . . 3  
2  cB  . . 3  
3  1, 2  wceq 1395  . 2 
4  vx  . . . . . 6  
5  4  cv 1394  . . . . 5 
6  5, 1  wcel 1818  . . . 4 
7  5, 2  wcel 1818  . . . 4 
8  6, 7  wb 184  . . 3 
9  8, 4  wal 1393  . 2 
10  3, 9  wb 184  1 
Colors of variables: wff setvar class 
This definition is referenced by: dfcleq 2450 
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