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Theorem mreexd 15499
 Description: In a Moore system, the closure operator is said to have the exchange property if, for all elements and of the base set and subsets of the base set such that is in the closure of but not in the closure of , is in the closure of (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mreexd.1
mreexd.2
mreexd.3
mreexd.4
mreexd.5
mreexd.6
Assertion
Ref Expression
mreexd
Distinct variable groups:   ,,   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   ()

Proof of Theorem mreexd
StepHypRef Expression
1 mreexd.2 . 2
2 mreexd.3 . . . 4
3 mreexd.1 . . . . 5
4 elpw2g 4588 . . . . 5
53, 4syl 17 . . . 4
62, 5mpbird 235 . . 3
7 mreexd.4 . . . . 5
87adantr 466 . . . 4
9 mreexd.5 . . . . . . . 8
109ad2antrr 730 . . . . . . 7
11 simplr 760 . . . . . . . . 9
12 simpr 462 . . . . . . . . . 10
1312sneqd 4014 . . . . . . . . 9
1411, 13uneq12d 3627 . . . . . . . 8
1514fveq2d 5885 . . . . . . 7
1610, 15eleqtrrd 2520 . . . . . 6
17 mreexd.6 . . . . . . . 8
1817ad2antrr 730 . . . . . . 7
1911fveq2d 5885 . . . . . . 7
2018, 19neleqtrrd 2542 . . . . . 6
2116, 20eldifd 3453 . . . . 5
22 simplr 760 . . . . . 6
23 simpllr 767 . . . . . . . 8
24 simpr 462 . . . . . . . . 9
2524sneqd 4014 . . . . . . . 8
2623, 25uneq12d 3627 . . . . . . 7
2726fveq2d 5885 . . . . . 6
2822, 27eleq12d 2511 . . . . 5
2921, 28rspcdv 3191 . . . 4
308, 29rspcimdv 3189 . . 3
316, 30rspcimdv 3189 . 2
321, 31mpd 15 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wa 370   wceq 1437   wcel 1870  wral 2782   cdif 3439   cun 3440   wss 3442  cpw 3985  csn 4002  cfv 5601 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609 This theorem is referenced by:  mreexmrid  15500
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