Theorem cf0 3705

Description: Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102.

Assertion

Ref	Expression
cf0	⊢ (cf ‘∅) = ∅

Proof of Theorem cf0

Step	Hyp	Ref	Expression
1		cfub 3703	. . 3 ⊢ (cf ‘∅) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y))}
2		0ss 1725	. . . . . . . . . . . . 13 ⊢ ∅ ⊆ ∪y
3	2	biantru 543	. . . . . . . . . . . 12 ⊢ (y ⊆ ∅ ↔ (y ⊆ ∅ ∧ ∅ ⊆ ∪y))
4		ss0b 1726	. . . . . . . . . . . 12 ⊢ (y ⊆ ∅ ↔ y = ∅)
5	3, 4	bitr3 153	. . . . . . . . . . 11 ⊢ ((y ⊆ ∅ ∧ ∅ ⊆ ∪y) ↔ y = ∅)
6	5	anbi2i 367	. . . . . . . . . 10 ⊢ ((x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y)) ↔ (x = (card ‘y) ∧ y = ∅))
7		ancom 333	. . . . . . . . . 10 ⊢ ((x = (card ‘y) ∧ y = ∅) ↔ (y = ∅ ∧ x = (card ‘y)))
8	6, 7	bitr 151	. . . . . . . . 9 ⊢ ((x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y)) ↔ (y = ∅ ∧ x = (card ‘y)))
9	8	biex 733	. . . . . . . 8 ⊢ (∃y(x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y)) ↔ ∃y(y = ∅ ∧ x = (card ‘y)))
10		0ex 1745	. . . . . . . . 9 ⊢ ∅ ∈ V
11		fveq2 2832	. . . . . . . . . 10 ⊢ (y = ∅ → (card ‘y) = (card ‘∅))
12	11	cleq2d 1112	. . . . . . . . 9 ⊢ (y = ∅ → (x = (card ‘y) ↔ x = (card ‘∅)))
13	10, 12	ceqsexv 1371	. . . . . . . 8 ⊢ (∃y(y = ∅ ∧ x = (card ‘y)) ↔ x = (card ‘∅))
14		card0 3630	. . . . . . . . 9 ⊢ (card ‘∅) = ∅
15	14	cleq2i 1111	. . . . . . . 8 ⊢ (x = (card ‘∅) ↔ x = ∅)
16	9, 13, 15	3bitr 155	. . . . . . 7 ⊢ (∃y(x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y)) ↔ x = ∅)
17	16	biabi 1181	. . . . . 6 ⊢ {x∣∃y(x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y))} = {x∣x = ∅}
18		df-sn 1811	. . . . . 6 ⊢ {∅} = {x∣x = ∅}
19	17, 18	eqtr4 1122	. . . . 5 ⊢ {x∣∃y(x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y))} = {∅}
20	19	inteqi 1969	. . . 4 ⊢ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y))} = ∩{∅}
21	10	intsn 1991	. . . 4 ⊢ ∩{∅} = ∅
22	20, 21	eqtr 1119	. . 3 ⊢ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ ∅ ∧ ∅ ⊆ ∪y))} = ∅
23	1, 22	sseqtr 1532	. 2 ⊢ (cf ‘∅) ⊆ ∅
24		ss0b 1726	. 2 ⊢ ((cf ‘∅) ⊆ ∅ ↔ (cf ‘∅) = ∅)
25	23, 24	mpbi 164	1 ⊢ (cf ‘∅) = ∅

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ⊆ wss 1487  ∅c0 1707  {csn 1808  ∪cuni 1919  ∩cint 1965   ‘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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