Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfval Structured version   Unicode version

Theorem cfval 8129
 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 8128 . . 3
2 intexab 4360 . . 3
31, 2sylib 190 . 2
4 sseq2 3372 . . . . . . . 8
5 raleq 2906 . . . . . . . 8
64, 5anbi12d 693 . . . . . . 7
76anbi2d 686 . . . . . 6
87exbidv 1637 . . . . 5
98abbidv 2552 . . . 4
109inteqd 4057 . . 3
11 df-cf 7830 . . 3
1210, 11fvmptg 5806 . 2
133, 12mpdan 651 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360  wex 1551   wceq 1653   wcel 1726  cab 2424  wral 2707  wrex 2708  cvv 2958   wss 3322  cint 4052  con0 4583  cfv 5456  ccrd 7824  ccf 7826 This theorem is referenced by:  cfub  8131  cflm  8132  cardcf  8134  cflecard  8135  cfeq0  8138  cfsuc  8139  cff1  8140  cfflb  8141  cfval2  8142  cflim3  8144 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-cf 7830
 Copyright terms: Public domain W3C validator