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Theorem isnacs 26749
 Description: Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f mrCls
Assertion
Ref Expression
isnacs NoeACS ACS
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem isnacs
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5750 . 2 NoeACS
2 elfvex 5750 . . 3 ACS
32adantr 452 . 2 ACS
4 fveq2 5720 . . . . . 6 ACS ACS
5 pweq 3794 . . . . . . . . 9
65ineq1d 3533 . . . . . . . 8
76rexeqdv 2903 . . . . . . 7 mrCls mrCls
87ralbidv 2717 . . . . . 6 mrCls mrCls
94, 8rabeqbidv 2943 . . . . 5 ACS mrCls ACS mrCls
10 df-nacs 26748 . . . . 5 NoeACS ACS mrCls
11 fvex 5734 . . . . . 6 ACS
1211rabex 4346 . . . . 5 ACS mrCls
139, 10, 12fvmpt 5798 . . . 4 NoeACS ACS mrCls
1413eleq2d 2502 . . 3 NoeACS ACS mrCls
15 fveq2 5720 . . . . . . . . 9 mrCls mrCls
16 isnacs.f . . . . . . . . 9 mrCls
1715, 16syl6eqr 2485 . . . . . . . 8 mrCls
1817fveq1d 5722 . . . . . . 7 mrCls
1918eqeq2d 2446 . . . . . 6 mrCls
2019rexbidv 2718 . . . . 5 mrCls
2120raleqbi1dv 2904 . . . 4 mrCls
2221elrab 3084 . . 3 ACS mrCls ACS
2314, 22syl6bb 253 . 2 NoeACS ACS
241, 3, 23pm5.21nii 343 1 NoeACS ACS
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  crab 2701  cvv 2948   cin 3311  cpw 3791  cfv 5446  cfn 7101  mrClscmrc 13800  ACScacs 13802  NoeACScnacs 26747 This theorem is referenced by:  nacsfg  26750  isnacs2  26751  isnacs3  26755  islnr3  27287 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-nacs 26748
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