Theorem cfle 3708

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 3629	. . 3 ⊢ (A ∈ On → (card ‘A) ⊆ A)
2		cflecard 3707	. . . 4 ⊢ (cf ‘A) ⊆ (card ‘A)
3		sstr2 1510	. . . 4 ⊢ ((cf ‘A) ⊆ (card ‘A) → ((card ‘A) ⊆ A → (cf ‘A) ⊆ A))
4	2, 3	ax-mp 6	. . 3 ⊢ ((card ‘A) ⊆ A → (cf ‘A) ⊆ A)
5	1, 4	syl 12	. 2 ⊢ (A ∈ On → (cf ‘A) ⊆ A)
6		0ss 1725	. . 3 ⊢ ∅ ⊆ A
7		cffnon 3702	. . . . . . . 8 ⊢ cf Fn On
8		fndm 2723	. . . . . . . 8 ⊢ (cf Fn On → dom cf = On)
9	7, 8	ax-mp 6	. . . . . . 7 ⊢ dom cf = On
10	9	eleq2i 1153	. . . . . 6 ⊢ (A ∈ dom cf ↔ A ∈ On)
11	10	negbii 162	. . . . 5 ⊢ (¬ A ∈ dom cf ↔ ¬ A ∈ On)
12		ndmfv 2848	. . . . 5 ⊢ (¬ A ∈ dom cf → (cf ‘A) = ∅)
13	11, 12	sylbir 176	. . . 4 ⊢ (¬ A ∈ On → (cf ‘A) = ∅)
14	13	sseq1d 1527	. . 3 ⊢ (¬ A ∈ On → ((cf ‘A) ⊆ A ↔ ∅ ⊆ A))
15	6, 14	mpbiri 169	. 2 ⊢ (¬ A ∈ On → (cf ‘A) ⊆ A)
16	5, 15	pm2.61i 110	1 ⊢ (cf ‘A) ⊆ A

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∅c0 1707  Oncon0 2199  dom cdm 2410   Fn wfn 2417   ‘cfv 2422  cardccrd 3620  cfccf 3622

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274  df-card 3623  df-cf 3625

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