Theorem cfom 3710

Description: Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102.

Assertion

Ref	Expression
cfom	⊢ (cf ‘ω) = ω

Proof of Theorem cfom

Step	Hyp	Ref	Expression
1		cflecard 3707	. . 3 ⊢ (cf ‘ω) ⊆ (card ‘ω)
2		cardom 3632	. . 3 ⊢ (card ‘ω) = ω
3	1, 2	sseqtr 1532	. 2 ⊢ (cf ‘ω) ⊆ ω
4		omex 3475	. . . 4 ⊢ ω ∈ V
5	4	intsn 1991	. . 3 ⊢ ∩{ω} = ω
6		cleqtr 1118	. . . . . . . . 9 ⊢ ((x = (card ‘y) ∧ (card ‘y) = ω) → x = ω)
7		visset 1350	. . . . . . . . . . . 12 ⊢ y ∈ V
8	7	unbnn2 3436	. . . . . . . . . . 11 ⊢ ((y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w) → y ≈ ω)
9		carden 3638	. . . . . . . . . . . 12 ⊢ ((y ∈ V ∧ ω ∈ V) → ((card ‘y) = (card ‘ω) ↔ y ≈ ω))
10	7, 4, 9	mp2an 520	. . . . . . . . . . 11 ⊢ ((card ‘y) = (card ‘ω) ↔ y ≈ ω)
11	8, 10	sylibr 175	. . . . . . . . . 10 ⊢ ((y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w) → (card ‘y) = (card ‘ω))
12	11, 2	syl6eq 1140	. . . . . . . . 9 ⊢ ((y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w) → (card ‘y) = ω)
13	6, 12	sylan2 346	. . . . . . . 8 ⊢ ((x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w)) → x = ω)
14	13	19.23aiv 952	. . . . . . 7 ⊢ (∃y(x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w)) → x = ω)
15	14	ss2abi 1552	. . . . . 6 ⊢ {x∣∃y(x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w))} ⊆ {x∣x = ω}
16		df-sn 1811	. . . . . 6 ⊢ {ω} = {x∣x = ω}
17	15, 16	sseqtr4 1533	. . . . 5 ⊢ {x∣∃y(x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w))} ⊆ {ω}
18		intss 1983	. . . . 5 ⊢ ({x∣∃y(x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w))} ⊆ {ω} → ∩{ω} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w))})
19	17, 18	ax-mp 6	. . . 4 ⊢ ∩{ω} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w))}
20		omelon 3476	. . . . 5 ⊢ ω ∈ On
21		cfval 3701	. . . . 5 ⊢ (ω ∈ On → (cf ‘ω) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w))})
22	20, 21	ax-mp 6	. . . 4 ⊢ (cf ‘ω) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ ω ∧ ∀z ∈ ω ∃w ∈ y z ⊆ w))}
23	19, 22	sseqtr4 1533	. . 3 ⊢ ∩{ω} ⊆ (cf ‘ω)
24	5, 23	eqsstr3 1531	. 2 ⊢ ω ⊆ (cf ‘ω)
25	3, 24	eqssi 1517	1 ⊢ (cf ‘ω) = ω

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  {csn 1808  ∩cint 1965   class class class wbr 2054  Oncon0 2199  ωcom 2372   ‘cfv 2422   ≈ cen 3271  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  ax-reg 1078  ax-inf 1079  ax-ac 1080

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-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  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-lim 2204  df-suc 2205  df-om 2373  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-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-cf 3625

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