Theorem cfub 3703

Description: An upper bound on cofinality.

Assertion

Ref	Expression
cfub	⊢ (cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))}

Distinct variable group(s):   x,y,A

Proof of Theorem cfub

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		ssel 1502	. . . . . . . . . . . . . . . . . 18 ⊢ (y ⊆ A → (w ∈ y → w ∈ A))
3		onelon 2223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((A ∈ On ∧ w ∈ A) → w ∈ On)
4	3	exp 291	. . . . . . . . . . . . . . . . . 18 ⊢ (A ∈ On → (w ∈ A → w ∈ On))
5	2, 4	sylan9r 360	. . . . . . . . . . . . . . . . 17 ⊢ ((A ∈ On ∧ y ⊆ A) → (w ∈ y → w ∈ On))
6		onelsst 2255	. . . . . . . . . . . . . . . . 17 ⊢ (w ∈ On → (z ∈ w → z ⊆ w))
7	5, 6	syl6 23	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ On ∧ y ⊆ A) → (w ∈ y → (z ∈ w → z ⊆ w)))
8	7	imdistand 342	. . . . . . . . . . . . . . 15 ⊢ ((A ∈ On ∧ y ⊆ A) → ((w ∈ y ∧ w) → (w ∈ y ∧ z ⊆ w)))
9	8	ancomsd 335	. . . . . . . . . . . . . 14 ⊢ ((A ∈ On ∧ y ⊆ A) → ((z ∈ w ∧ w ∈ y) → (w ∈ y ∧ z ⊆ w)))
10	9	19.22dv 947	. . . . . . . . . . . . 13 ⊢ ((A ∈ On ∧ y ⊆ A) → (∃w(z ∈ w ∧ w ∈ y) → ∃w(w ∈ y ∧ z ⊆ w)))
11		eluni 1922	. . . . . . . . . . . . 13 ⊢ (z ∈ ∪y ↔ ∃w(z ∈ w ∧ w ∈ y))
12		df-rex 1206	. . . . . . . . . . . . 13 ⊢ (∃w ∈ y z ⊆ w ↔ ∃w(w ∈ y ∧ z ⊆ w))
13	10, 11, 12	3imtr4g 426	. . . . . . . . . . . 12 ⊢ ((A ∈ On ∧ y ⊆ A) → (z ∈ ∪y → ∃w ∈ y z ⊆ w))
14	13	r19.20sdv 1257	. . . . . . . . . . 11 ⊢ ((A ∈ On ∧ y ⊆ A) → (∀z ∈ A z ∈ ∪y → ∀z ∈ A ∃w ∈ y z ⊆ w))
15		dfss3 1498	. . . . . . . . . . 11 ⊢ (A ⊆ ∪y ↔ ∀z ∈ A z ∈ ∪y)
16	14, 15	syl5ib 181	. . . . . . . . . 10 ⊢ ((A ∈ On ∧ y ⊆ A) → (A ⊆ ∪y → ∀z ∈ A ∃w ∈ y z ⊆ w))
17	16	exp 291	. . . . . . . . 9 ⊢ (A ∈ On → (y ⊆ A → (A ⊆ ∪y → ∀z ∈ A ∃w ∈ y z ⊆ w)))
18	17	imdistand 342	. . . . . . . 8 ⊢ (A ∈ On → ((y ⊆ A ∧ A ⊆ ∪y) → (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)))
19	18	anim2d 433	. . . . . . 7 ⊢ (A ∈ On → ((x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y)) → (x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
20	19	19.22dv 947	. . . . . 6 ⊢ (A ∈ On → (∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y)) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
21	20	19.21aiv 943	. . . . 5 ⊢ (A ∈ On → ∀x(∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y)) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
22		ss2ab 1551	. . . . 5 ⊢ ({x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ↔ ∀x(∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y)) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))))
23	21, 22	sylibr 175	. . . 4 ⊢ (A ∈ On → {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
24		intss 1983	. . . 4 ⊢ ({x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))} ⊆ {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))} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))})
25	23, 24	sylfnbsp;12	. . 3 ⊢ (A ∈ On → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))})
26	1, 25	eqsstrd 1534	. 2 ⊢ (A ∈ On → (cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))})
27		0ss 1725	. . 3 ⊢ ∅ ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))}
28		cffnon 3702	. . . . . . . 8 ⊢ cf Fn On
29		fndm 2723	. . . . . . . 8 ⊢ (cf Fn On → dom cf = On)
30	28, 29	ax-mp 6	. . . . . . 7 ⊢ dom cf = On
31	30	eleq2i 1153	. . . . . 6 ⊢ (A ∈ dom cf ↔ A ∈ On)
32	31	negbii 162	. . . . 5 ⊢ (¬ A ∈ dom cf ↔ ¬ A ∈ On)
33		ndmfv 2848	. . . . 5 ⊢ (¬ A ∈ dom cf → (cf ‘A) = ∅)
34	32, 33	sylbir 176	. . . 4 ⊢ (¬ A ∈ On → (cf ‘A) = ∅)
35	34	sseq1d 1527	. . 3 ⊢ (¬ A ∈ On → ((cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))} ↔ ∅ ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))}))
36	27, 35	mpbiri 169	. 2 ⊢ (¬ A ∈ On → (cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))})
37	26, 36	pm2.61i 110	1 ⊢ (cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))}

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919  ∩cint 1965  Oncon0 2199  dom cdm 2410   Fn wfn 2417   ‘cfv 2422  cardccrd 3620  cfccf 3622

This theorem is referenced by:  cflim 3704  cf0 3705

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-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-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-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-fv 2438  df-cf 3625

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