Theorem cardval 3633

Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 3661 for a simpler version of its value.

Assertion

Ref	Expression
cardval	⊢ (card ‘A) = ∩{x ∈ On∣x ≈ A}

Distinct variable group(s):   x,A

Proof of Theorem cardval

Step	Hyp	Ref	Expression
1		numth2 3600	. . . . 5 ⊢ ∃x ∈ On x ≈ A
2		intexrab 1988	. . . . 5 ⊢ (∃x ∈ On x ≈ A ↔ ∩{x ∈ On∣x ≈ A} ∈ V)
3	1, 2	mpbi 164	. . . 4 ⊢ ∩{x ∈ On∣x ≈ A} ∈ V
4		breq2 2066	. . . . . . 7 ⊢ (y = A → (x ≈ y ↔ x ≈ A))
5	4	birabsdv 1344	. . . . . 6 ⊢ (y = A → {x ∈ On∣x ≈ y} = {x ∈ On∣x ≈ A})
6	5	inteqd 1970	. . . . 5 ⊢ (y = A → ∩{x ∈ On∣x ≈ y} = ∩{x ∈ On∣x ≈ A})
7	6	fvopabg 2872	. . . 4 ⊢ ((A ∈ V ∧ ∩{x ∈ On∣x ≈ A} ∈ V) → ({⟨y, z⟩∣z = ∩{x ∈ On∣x ≈ y}} ‘A) = ∩{x ∈ On∣x ≈ A})
8	3, 7	mpan2 519	. . 3 ⊢ (A ∈ V → ({⟨y, z⟩∣z = ∩{x ∈ On∣x ≈ y}} ‘A) = ∩{x ∈ On∣x ≈ A})
9		df-card 3623	. . . 4 ⊢ card = {⟨y, z⟩∣z = ∩{x ∈ On∣x ≈ y}}
10	9	fveq1i 2833	. . 3 ⊢ (card ‘A) = ({⟨y, z⟩∣z = ∩{x ∈ On∣x ≈ y}} ‘A)
11	8, 10	syl5eq 1136	. 2 ⊢ (A ∈ V → (card ‘A) = ∩{x ∈ On∣x ≈ A})
12		fvprc 2829	. . 3 ⊢ (¬ A ∈ V → (card ‘A) = ∅)
13		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
14	13	enref 3295	. . . . . . . . . 10 ⊢ x ≈ x
15		brprc 2097	. . . . . . . . . 10 ⊢ (¬ A ∈ V → (x ≈ A ↔ x ≈ x))
16	14, 15	mpbiri 169	. . . . . . . . 9 ⊢ (¬ A ∈ V → x ≈ A)
17	16	biantrud&nbs(;545	. . . . . . . 8 ⊢ (¬ A ∈ V → (x ∈ On ↔ (x ∈ On ∧ x ≈ A)))
18	17	biabdv 1183	. . . . . . 7 ⊢ (¬ A ∈ V → {x∣x ∈ On} = {x∣(x ∈ On ∧ x ≈ A)})
19		df-rab 1208	. . . . . . 7 ⊢ {x ∈ On∣x ≈ A} = {x∣(x ∈ On ∧ x ≈ A)}
20	18, 19	syl6reqr 1143	. . . . . 6 ⊢ (¬ A ∈ V → {x ∈ On∣x ≈ A} = {x∣x ∈ On})
21		abid2 1186	. . . . . 6 ⊢ {x∣x ∈ On} = On
22	20, 21	syl6eq 1140	. . . . 5 ⊢ (¬ A ∈ V → {x ∈ On∣x ≈ A} = On)
23	22	inteqd 1970	. . . 4 ⊢ (¬ A ∈ V → ∩{x ∈ On∣x ≈ A} = ∩On)
24		inton 2281	. . . 4 ⊢ ∩On = ∅
25	23, 24	syl6eq 1140	. . 3 ⊢ (¬ A ∈ V → ∩{x ∈ On∣x ≈ A} = ∅)
26	12, 25	eqtr4d 1131	. 2 ⊢ (¬ A ∈ V → (card ‘A) = ∩{x ∈ On∣x ≈ A})
27	11, 26	pm2.61i 110	1 ⊢ (card ‘A) = ∩{x ∈ On∣x ≈ A}

Syntax hints:  ¬ wn 1   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  Vcvv 1348  ∅c0 1707  ∩cint 1965   class class class wbr 2054  {copab 2055  Oncon0 2199   ‘cfv 2422   ≈ cen 3271  cardccrd 3620

This theorem is referenced by:  cardon 3634  cardid 3635  oncard 3636  cardne 3637  iscard2 3660

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-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-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-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-suc 2205  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

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