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Theorem nacsfg 26750
 Description: In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f mrCls
Assertion
Ref Expression
nacsfg NoeACS
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem nacsfg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isnacs.f . . . . 5 mrCls
21isnacs 26749 . . . 4 NoeACS ACS
32simprbi 451 . . 3 NoeACS
4 eqeq1 2441 . . . . 5
54rexbidv 2718 . . . 4
65rspcva 3042 . . 3
73, 6sylan2 461 . 2 NoeACS
87ancoms 440 1 NoeACS
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698   cin 3311  cpw 3791  cfv 5446  cfn 7101  mrClscmrc 13800  ACScacs 13802  NoeACScnacs 26747 This theorem is referenced by:  isnacs3  26755 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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