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Theorem fin23lem7 8717
Description: Lemma for isfin2-2 8720. 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:   ,   ,

Proof of Theorem fin23lem7
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 n0 3794 . . . 4
2 difss 3630 . . . . . . . 8
3 elpw2g 4615 . . . . . . . . 9
43ad2antrr 725 . . . . . . . 8
52, 4mpbiri 233 . . . . . . 7
6 simpr 461 . . . . . . . . . . 11
76sselda 3503 . . . . . . . . . 10
87elpwid 4022 . . . . . . . . 9
9 dfss4 3731 . . . . . . . . 9
108, 9sylib 196 . . . . . . . 8
11 simpr 461 . . . . . . . 8
1210, 11eqeltrd 2545 . . . . . . 7
13 difeq2 3615 . . . . . . . . 9
1413eleq1d 2526 . . . . . . . 8
1514rspcev 3210 . . . . . . 7
165, 12, 15syl2anc 661 . . . . . 6
1716ex 434 . . . . 5
1817exlimdv 1724 . . . 4
191, 18syl5bi 217 . . 3
20193impia 1193 . 2
21 rabn0 3805 . 2
2220, 21sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652  E.wrex 2808  {crab 2811  \cdif 3472  C_wss 3475   c0 3784  ~Pcpw 4012
This theorem is referenced by:  fin2i2  8719  isfin2-2  8720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-pw 4014
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