Theorem cfle 4932

Description: Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102.

Assertion

Ref	Expression
cfle	|- (cf` A) (_ A

Proof of Theorem cfle

Step	Hyp	Ref	Expression
1		cardonle 4839	. . 3 |- (A e. On -> (card` A) (_ A)
2		cflecard 4931	. . . 4 |- (cf` A) (_ (card` A)
3		sstr2 2075	. . . 4 |- ((cf` A) (_ (card` A) -> ((card` A) (_ A -> (cf` A) (_ A))
4	2, 3	ax-mp 7	. . 3 |- ((card` A) (_ A -> (cf` A) (_ A)
5	1, 4	syl 10	. 2 |- (A e. On -> (cf` A) (_ A)
6		0ss 2306	. . 3 |- (/) (_ A
7		cffnon 4926	. . . . . . . 8 |- cf Fn On
8		fndm 3594	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
9	7, 8	ax-mp 7	. . . . . . 7 |- dom cf = On
10	9	eleq2i 1541	. . . . . 6 |- (A e. dom cf <-> A e. On)
11	10	negbii 187	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
12		ndmfv 3752	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
13	11, 12	sylbir 201	. . . 4 |- (-. A e. On -> (cf` A) = (/))
14	13	sseq1d 2092	. . 3 |- (-. A e. On -> ((cf` A) (_ A <-> (/) (_ A))
15	6, 14	mpbiri 194	. 2 |- (-. A e. On -> (cf` A) (_ A)
16	5, 15	pm2.61i 126	1 |- (cf` A) (_ A

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   e. wcel 960   (_ wss 2051  (/)c0 2284  Oncon0 2955  dom cdm 3177   Fn wfn 3184  ` cfv 3189  cardccrd 4830  cfccf 4832

This theorem is referenced by:  cfom 4935

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-pr 2786  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-int 2539  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-id 2842  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-f 3201  df-f1 3202  df-fo 3203  df-f1o 3204  df-fv 3205  df-en 4375  df-card 4833  df-cf 4835

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