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Theorem zfreg 8115
Description: The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form." There is also a "strong form," not requiring that  A be a set, that can be proved with more difficulty (see zfregs 8221). (Contributed by NM, 26-Nov-1995.)
Hypothesis
Ref Expression
zfreg.1  |-  A  e. 
_V
Assertion
Ref Expression
zfreg  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
Distinct variable group:    x, A

Proof of Theorem zfreg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 zfreg.1 . . 3  |-  A  e. 
_V
21zfregcl 8114 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3743 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disj 3807 . . 3  |-  ( ( x  i^i  A )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2891 . 2  |-  ( E. x  e.  A  ( x  i^i  A )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 270 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1446   E.wex 1665    e. wcel 1889    =/= wne 2624   A.wral 2739   E.wrex 2740   _Vcvv 3047    i^i cin 3405   (/)c0 3733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-reg 8112
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-v 3049  df-dif 3409  df-in 3413  df-nul 3734
This theorem is referenced by:  en3lp  8126  inf3lem3  8140  setindtr  35891
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