Theorem cflecard 3707

Description: Cofinality is bounded by the cardinality of its argument.

Assertion

Ref	Expression
cflecard	⊢ (cf ‘A) ⊆ (card ‘A)

Proof of Theorem cflecard

Step	Hyp	Ref	Expression
1		cfval 3701	. . 3 ⊢ (A ∈ On → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
2		ssid 1519	. . . . . . . . . 10 ⊢ A ⊆ A
3		ssid 1519	. . . . . . . . . . . 12 ⊢ z ⊆ z
4		sseq2 1522	. . . . . . . . . . . . 13 ⊢ (w = z → (z ⊆ w ↔ z ⊆ z))
5	4	rcla4ev 1403	. . . . . . . . . . . 12 ⊢ ((z ∈ A ∧ z ⊆ z) → ∃w ∈ A z ⊆ w)
6	3, 5	mpan2 519	. . . . . . . . . . 11 ⊢ (z ∈ A → ∃w ∈ A z ⊆ w)
7	6	rgen 1247	. . . . . . . . . 10 ⊢ ∀z ∈ A ∃w ∈ A z ⊆ w
8	2, 7	pm3.2i 234	. . . . . . . . 9 ⊢ (A ⊆ A ∧ ∀z ∈ A ∃w ∈ A z ⊆ w)
9		fveq2 2832	. . . . . . . . . . . 12 ⊢ (y = A → (card ‘y) = (card ‘A))
10	9	cleq2d 1112	. . . . . . . . . . 11 ⊢ (y = A → (x = (card ‘y) ↔ x = (card ‘A)))
11		sseq1 1521	. . . . . . . . . . . 12 ⊢ (y = A → (y ⊆ A ↔ A ⊆ A))
12		rexeq 1325	. . . . . . . . . . . . 13 ⊢ (y = A → (∃w ∈ y z ⊆ w ↔ ∃w ∈ A z ⊆ w))
13	12	biraldv 1219	. . . . . . . . . . . 12 ⊢ (y = A → (∀z ∈ A ∃w ∈ y z ⊆ w ↔ ∀z ∈ A ∃w ∈ A z ⊆ w))
14	11, 13	anbi12d 476	. . . . . . . . . . 11 ⊢ (y = A → ((y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w) ↔ (A ⊆ A ∧ ∀z ∈ A ∃w ∈ A z ⊆ w)))
15	10, 14	anbi12d 476	. . . . . . . . . 10 ⊢ (y = A → ((x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)) ↔ (x = (card ‘A) ∧ (A ⊆ A ∧ ∀z ∈ A ∃w ∈ A z ⊆ w))))
16	15	cla4egv 1397	. . . . . . . . 9 ⊢ (A ∈ On → ((x = (card ‘A) ∧ (A ⊆ A ∧ ∀z ∈ A ∃w ∈ A z ⊆ w)) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
17	8, 16	mpan2i 522	. . . . . . . 8 ⊢ (A ∈ On → (x = (card ‘A) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
18	17	19.21aiv 943	. . . . . . 7 ⊢ (A ∈ On → ∀x(x = (card ‘A) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
19		ss2ab 1551	. . . . . . 7 ⊢ ({x∣x = (card ‘A)} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ↔ ∀x(x = (card ‘A) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
20	18, 19	sylibr 175	. . . . . 6 ⊢ (A ∈ On → {x∣x = (card ‘A)} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
21		df-sn 1811	. . . . . 6 ⊢ {(card ‘A)} = {x∣x = (card ‘A)}
22	20, 21	syl5ss 1544	. . . . 5 ⊢ (A ∈ On → {(card ‘A)} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
23		intss 1983	. . . . 5 ⊢ ({(card ‘A)} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ⊆ ∩{(card ‘A)})
24	22, 23	syl 12	. . . 4 ⊢ (A ∈ On → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ⊆ ∩{(card ‘A)})
25		fvex 2838	. . . . 5 ⊢ (card ‘A) ∈ V
26	25	intsn 1991	. . . 4 ⊢ ∩{(card ‘A)} = (card ‘A)
27	24, 26	syl6ss 1546	. . 3 ⊢ (A ∈ On → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ⊆ (card ‘A))
28	1, 27	eqsstrd 1534	. 2 ⊢ (A ∈ On → (cf ‘A) ⊆ (card ‘A))
29		0ss 1725	. . 3 ⊢ ∅ ⊆ (card ‘A)
30		cffnon 3702	. . . . . . . 8 ⊢ cf Fn On
31		fndm 2723	. . . . . . . 8 ⊢ (cf Fn On → dom cf = On)
32	30, 31	ax-mp 6	. . . . . . 7 ⊢ dom cf = On
33	32	eleq2i 1153	. . . . . 6 ⊢ (A ∈ dom cf ↔ A ∈ On)
34	33	negbii 162	. . . . 5 ⊢ (¬ A ∈ dom cf ↔ ¬ A ∈ On)
35		ndmfv 2848	. . . . 5 ⊢ (¬ A ∈ dom cf → (cf ‘A) = ∅)
36	34, 35	sylbir 176	. . . 4 ⊢ (¬ A ∈ On → (cf ‘A) = ∅)
37	36	sseq1d 1527	. . 3 ⊢ (¬ A ∈ On → ((cf ‘A) ⊆ (card ‘A) ↔ ∅ ⊆ (card ‘A)))
38	29, 37	mpbiri 169	. 2 ⊢ (¬ A ∈ On → (cf ‘A) ⊆ (card ‘A))
39	28, 38	pm2.61i 110	1 ⊢ (cf ‘A) ⊆ (card ‘A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ∅c0 1707  {csn 1808  ∩cint 1965  Oncon0 2199  dom cdm 2410   Fn wfn 2417   ‘cfv 2422  cardccrd 3620  cfccf 3622

This theorem is referenced by:  cfle 3708  cfom 3710

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-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-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-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-id 2125  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-fv 2438  df-cf 3625

metamath.org