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Theorem cflem 7756
 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 3118 . . 3
2 ssid 3118 . . . . 5
3 sseq2 3121 . . . . . 6
43rcla4ev 2821 . . . . 5
52, 4mpan2 655 . . . 4
65rgen 2570 . . 3
7 sseq1 3120 . . . . 5
8 rexeq 2690 . . . . . 6
98ralbidv 2527 . . . . 5
107, 9anbi12d 694 . . . 4
1110cla4egv 2806 . . 3
121, 6, 11mp2ani 662 . 2
13 fvex 5391 . . . . . 6
1413isseti 2733 . . . . 5
15 19.41v 2034 . . . . 5
1614, 15mpbiran 889 . . . 4
1716exbii 1580 . . 3
18 excom 1765 . . 3
1917, 18bitr3i 244 . 2
2012, 19sylib 190 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360  wex 1537   wceq 1619   wcel 1621  wral 2509  wrex 2510   wss 3078  cfv 4592  ccrd 7452 This theorem is referenced by:  cfval  7757  cff  7758  cff1  7768 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-sn 3550  df-pr 3551  df-uni 3728  df-fv 4608
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