Theorem cfval 8648

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.

Step	Hyp	Ref	Expression
1		cflem 8647	. . 3
2		intexab 4610	. . 3
3	1, 2	sylib 196	. 2
4		sseq2 3525	. . . . . . . 8
5		raleq 3054	. . . . . . . 8
6	4, 5	anbi12d 710	. . . . . . 7
7	6	anbi2d 703	. . . . . 6
8	7	exbidv 1714	. . . . 5
9	8	abbidv 2593	. . . 4
10	9	inteqd 4291	. . 3
11		df-cf 8343	. . 3
12	10, 11	fvmptg 5954	. 2
13	3, 12	mpdan 668	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808  cvv 3109  C_wss 3475  |^|cint 4286  con0 4883  `cfv 5593  ccrd 8337  ccf 8339

This theorem is referenced by:  cfub  8650  cflm  8651  cardcf  8653  cflecard  8654  cfeq0  8657  cfsuc  8658  cff1  8659  cfflb  8660  cfval2  8661  cflim3  8663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-int 4287  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-cf 8343