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Theorem zfreg2 7194
Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7193) 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

Proof of Theorem zfreg2
StepHypRef Expression
1 zfreg2.1 . . 3  |-  A  e. 
_V
21zfregcl 7192 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3371 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disjr 3403 . . 3  |-  ( ( A  i^i  x )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2532 . 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 1537    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510   _Vcvv 2727    i^i cin 3077   (/)c0 3362
This theorem is referenced by:  zfregfr  7200
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-reg 7190
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-v 2729  df-dif 3081  df-in 3085  df-nul 3363
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