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Theorem cflem 8126
 Description: A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set . (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
cflem
Distinct variable group:   ,,,,
Allowed substitution hints:   (,,,)

Proof of Theorem cflem
StepHypRef Expression
1 ssid 3367 . . 3
2 ssid 3367 . . . . 5
3 sseq2 3370 . . . . . 6
43rspcev 3052 . . . . 5
52, 4mpan2 653 . . . 4
65rgen 2771 . . 3
7 sseq1 3369 . . . . 5
8 rexeq 2905 . . . . . 6
98ralbidv 2725 . . . . 5
107, 9anbi12d 692 . . . 4
1110spcegv 3037 . . 3
121, 6, 11mp2ani 660 . 2
13 fvex 5742 . . . . . 6
1413isseti 2962 . . . . 5
15 19.41v 1924 . . . . 5
1614, 15mpbiran 885 . . . 4
1716exbii 1592 . . 3
18 excom 1756 . . 3
1917, 18bitr3i 243 . 2
2012, 19sylib 189 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1550   wceq 1652   wcel 1725  wral 2705  wrex 2706   wss 3320  cfv 5454  ccrd 7822 This theorem is referenced by:  cfval  8127  cff  8128  cff1  8138 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-pr 3821  df-uni 4016  df-iota 5418  df-fv 5462
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