Theorem oncardval 3626

Description: The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 3633, this theorem does not require the Axiom of Choice.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem oncardval

Step	Hyp	Ref	Expression
1		enrefg 3294	. . . . . 6 ⊢ (A ∈ On → A ≈ A)
2		breq1 2065	. . . . . . 7 ⊢ (x = A → (x ≈ A ↔ A ≈ A))
3	2	rcla4ev 1403	. . . . . 6 ⊢ ((A ∈ On ∧ A ≈ A) → ∃x ∈ On x ≈ A)
4	1, 3	mpdan 527	. . . . 5 ⊢ (A ∈ On → ∃x ∈ On x ≈ A)
5		rabn0 1716	. . . . 5 ⊢ (¬ {x ∈ On∣x ≈ A} = ∅ ↔ ∃x ∈ On x ≈ A)
6	4, 5	sylibr 175	. . . 4 ⊢ (A ∈ On → ¬ {x ∈ On∣x ≈ A} = ∅)
7		ssrab 1556	. . . . 5 ⊢ {x ∈ On∣x ≈ A} ⊆ On
8		oninton 2267	. . . . 5 ⊢ (({x ∈ On∣x ≈ A} ⊆ On ∧ ¬ {x ∈ On∣x ≈ A} = ∅) → ∩{x ∈ On∣x ≈ A} ∈ On)
9	7, 8	mpan 518	. . . 4 ⊢ (¬ {x ∈ On∣x ≈ A} = ∅ → ∩{x ∈ On∣x ≈ A} ∈ On)
10	6, 9	syl 12	. . 3 ⊢ (A ∈ On → ∩{x ∈ On∣x ≈ A} ∈ On)
11		breq2 2066	. . . . . 6 ⊢ (y = A → (x ≈ y ↔ x ≈ A))
12	11	birabsdv 1344	. . . . 5 ⊢ (y = A → {x ∈ On∣x ≈ y} = {x ∈ On∣x ≈ A})
13	12	inteqd 1970	. . . 4 ⊢ (y = A → ∩{x ∈ On∣x ≈ y} = ∩{x ∈ On∣x ≈ A})
14	13	fvopabg 2872	. . 3 ⊢ ((A ∈ On ∧ ∩{x ∈ On∣x ≈ A} ∈ On) → ({⟨y, z⟩∣z = ∩{x ∈ On∣x ≈ y}} ‘A) = ∩{x ∈ On∣x ≈ A})
15	10, 14	mpdan 527	. 2 ⊢ (A ∈ On → ({⟨y, z⟩∣z = ∩{x ∈ On∣x ≈ y}} ‘A) = ∩{x ∈ On∣x ≈ A})
16		df-card 3623	. . 3 ⊢ card = {⟨y, z⟩∣z = ∩{x ∈ On∣x ≈ y}}
17	16	fveq1i 2833	. 2 ⊢ (card ‘A) = ({⟨y, z⟩∣z = ∩{x ∈ On∣x ≈ y}} ‘A)
18	15, 17	syl5eq 1136	1 ⊢ (A ∈ On → (card ‘A) = ∩{x ∈ On∣x ≈ A})

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204   ⊆ wss 1487  ∅c0 1707  ∩cint 1965   class class class wbr 2054  {copab 2055  Oncon0 2199   ‘cfv 2422   ≈ cen 3271  cardccrd 3620

This theorem is referenced by:  oncardon 3627  oncardid 3628  cardonle 3629

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

metamath.org