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Theorem fin23lem12 8766
 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 7179 . 2
3 fvex 5880 . . 3
4 fveq2 5870 . . . . . . 7
54ineq1d 3635 . . . . . 6
65eqeq1d 2455 . . . . 5
76, 5ifbieq2d 3908 . . . 4
8 ineq2 3630 . . . . . 6
98eqeq1d 2455 . . . . 5
10 id 22 . . . . 5
119, 10, 8ifbieq12d 3910 . . . 4
12 eqid 2453 . . . 4
133inex2 4548 . . . . 5
143, 13ifex 3951 . . . 4
157, 11, 12, 14ovmpt2 6437 . . 3
163, 15mpan2 678 . 2
172, 16eqtrd 2487 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1446   wcel 1889  cvv 3047   cin 3405  c0 3733  cif 3883  cuni 4201   crn 4838   csuc 5428  cfv 5585  (class class class)co 6295   cmpt2 6297  com 6697  seq𝜔cseqom 7169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-seqom 7170 This theorem is referenced by:  fin23lem13  8767  fin23lem14  8768  fin23lem19  8771
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