Theorem cvjust 2451

Description: Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1394, which allows us to substitute a setvar variable for a class variable. See also cab 2442 and df-clab 2443. Note that this is not a rigorous justification, because cv 1394 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.)

Assertion

Ref	Expression
cvjust

Distinct variable group:   ,

Proof of Theorem cvjust

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2450	. 2
2		df-clab 2443	. . 3
3		elsb3 2178	. . 3
4	2, 3	bitr2i 250	. 2
5	1, 4	mpgbir 1622	1

Syntax hints:  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818  {cab 2442

This theorem is referenced by:  cnambfre  30063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449