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Theorem fin23lem7 8201
 Description: Lemma for isfin2-2 8204. 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 3639 . . . 4
2 difss 3476 . . . . . . . 8
3 elpw2g 4366 . . . . . . . . 9
43ad2antrr 708 . . . . . . . 8
52, 4mpbiri 226 . . . . . . 7
6 simpr 449 . . . . . . . . . . 11
76sselda 3350 . . . . . . . . . 10
87elpwid 3810 . . . . . . . . 9
9 dfss4 3577 . . . . . . . . 9
108, 9sylib 190 . . . . . . . 8
11 simpr 449 . . . . . . . 8
1210, 11eqeltrd 2512 . . . . . . 7
13 difeq2 3461 . . . . . . . . 9
1413eleq1d 2504 . . . . . . . 8
1514rspcev 3054 . . . . . . 7
165, 12, 15syl2anc 644 . . . . . 6
1716ex 425 . . . . 5
1817exlimdv 1647 . . . 4
191, 18syl5bi 210 . . 3
20193impia 1151 . 2
21 rabn0 3649 . 2
2220, 21sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937  wex 1551   wceq 1653   wcel 1726   wne 2601  wrex 2708  crab 2711   cdif 3319   wss 3322  c0 3630  cpw 3801 This theorem is referenced by:  fin2i2  8203  isfin2-2  8204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803
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