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Theorem zfreg2 7305
Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7304) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.)
Hypothesis
Ref Expression
zfreg2.1  |-  A  e. 
_V
Assertion
Ref Expression
zfreg2  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
Distinct variable group:    x, A
Dummy variable  y is distinct from all other variables.

Proof of Theorem zfreg2
StepHypRef Expression
1 zfreg2.1 . . 3  |-  A  e. 
_V
21zfregcl 7303 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3465 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disjr 3497 . . 3  |-  ( ( A  i^i  x )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2569 . 2  |-  ( E. x  e.  A  ( A  i^i  x )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 259 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   E.wex 1529    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   _Vcvv 2789    i^i cin 3152   (/)c0 3456
This theorem is referenced by:  zfregfr  7311
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-reg 7301
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-v 2791  df-dif 3156  df-in 3160  df-nul 3457
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