Theorem cfle 4913

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 4822	. . 3 |- (A e. On -> (card` A) (_ A)
2		cflecard 4912	. . . 4 |- (cf` A) (_ (card` A)
3		sstr2 2071	. . . 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 2301	. . 3 |- (/) (_ A
7		cffnon 4907	. . . . . . . 8 |- cf Fn On
8		fndm 3587	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
9	7, 8	ax-mp 7	. . . . . . 7 |- dom cf = On
10	9	eleq2i 1538	. . . . . 6 |- (A e. dom cf <-> A e. On)
11	10	negbii 187	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
12		ndmfv 3745	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
13	11, 12	sylbir 201	. . . 4 |- (-. A e. On -> (cf` A) = (/))
14	13	sseq1d 2088	. . 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 956   e. wcel 958   (_ wss 2047  (/)c0 2280  Oncon0 2948  dom cdm 3170   Fn wfn 3177  ` cfv 3182  cardccrd 4813  cfccf 4815

This theorem is referenced by:  cfom 4916

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-en 4368  df-card 4816  df-cf 4818

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