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Theorem cfval 8083
 Description: Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number is the cardinality (size) of the smallest unbounded subset of the ordinal number. Unbounded means that for every member of , there is a member of that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cfval
Distinct variable group:   ,,,,

Proof of Theorem cfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cflem 8082 . . 3
2 intexab 4318 . . 3
31, 2sylib 189 . 2
4 sseq2 3330 . . . . . . . 8
5 raleq 2864 . . . . . . . 8
64, 5anbi12d 692 . . . . . . 7
76anbi2d 685 . . . . . 6
87exbidv 1633 . . . . 5
98abbidv 2518 . . . 4
109inteqd 4015 . . 3
11 df-cf 7784 . . 3
1210, 11fvmptg 5763 . 2
133, 12mpdan 650 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1547   wceq 1649   wcel 1721  cab 2390  wral 2666  wrex 2667  cvv 2916   wss 3280  cint 4010  con0 4541  cfv 5413  ccrd 7778  ccf 7780 This theorem is referenced by:  cfub  8085  cflm  8086  cardcf  8088  cflecard  8089  cfeq0  8092  cfsuc  8093  cff1  8094  cfflb  8095  cfval2  8096  cflim3  8098 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-cf 7784
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