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Theorem fin23lem12 8242
 Description: The beginning of the proof that every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets). This first section is dedicated to the construction of and its intersection. First, the value of at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
Ref Expression
fin23lem.a seq𝜔
Assertion
Ref Expression
fin23lem12
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   ()   ()

Proof of Theorem fin23lem12
StepHypRef Expression
1 fin23lem.a . . 3 seq𝜔
21seqomsuc 6743 . 2
3 fvex 5771 . . 3
4 fveq2 5757 . . . . . . 7
54ineq1d 3527 . . . . . 6
65eqeq1d 2450 . . . . 5
76, 5ifbieq2d 3783 . . . 4
8 ineq2 3522 . . . . . 6
98eqeq1d 2450 . . . . 5
10 id 21 . . . . 5
119, 10, 8ifbieq12d 3785 . . . 4
12 eqid 2442 . . . 4
133inex2 4374 . . . . 5
143, 13ifex 3821 . . . 4
157, 11, 12, 14ovmpt2 6238 . . 3
163, 15mpan2 654 . 2
172, 16eqtrd 2474 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1727  cvv 2962   cin 3305  c0 3613  cif 3763  cuni 4039   csuc 4612  com 4874   crn 4908  cfv 5483  (class class class)co 6110   cmpt2 6112  seq𝜔cseqom 6733 This theorem is referenced by:  fin23lem13  8243  fin23lem14  8244  fin23lem19  8247 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 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-recs 6662  df-rdg 6697  df-seqom 6734
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