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Theorem fin23lem7 8032
 Description: Lemma for isfin2-2 8035. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
fin23lem7
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem fin23lem7
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 n0 3540 . . . 4
2 difss 3379 . . . . . . . 8
3 elpw2g 4255 . . . . . . . . 9
43ad2antrr 706 . . . . . . . 8
52, 4mpbiri 224 . . . . . . 7
6 simpr 447 . . . . . . . . . . 11
76sselda 3256 . . . . . . . . . 10
8 elpwi 3709 . . . . . . . . . 10
97, 8syl 15 . . . . . . . . 9
10 dfss4 3479 . . . . . . . . 9
119, 10sylib 188 . . . . . . . 8
12 simpr 447 . . . . . . . 8
1311, 12eqeltrd 2432 . . . . . . 7
14 difeq2 3364 . . . . . . . . 9
1514eleq1d 2424 . . . . . . . 8
1615rspcev 2960 . . . . . . 7
175, 13, 16syl2anc 642 . . . . . 6
1817ex 423 . . . . 5
1918exlimdv 1636 . . . 4
201, 19syl5bi 208 . . 3
21203impia 1148 . 2
22 rabn0 3550 . 2
2321, 22sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3a 934  wex 1541   wceq 1642   wcel 1710   wne 2521  wrex 2620  crab 2623   cdif 3225   wss 3228  c0 3531  cpw 3701 This theorem is referenced by:  fin2i2  8034  isfin2-2  8035 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-in 3235  df-ss 3242  df-nul 3532  df-pw 3703
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