Theorem cardaleph 8473

Description: Given any transfinite cardinal number A, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
cardaleph	|- ( ( om C_ A /\ ( card ` A ) = A ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )

Distinct variable group:   x, A

Proof of Theorem cardaleph

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardon 8328	. . . . . . . . 9 |- ( card ` A ) e. On
2		eleq1 2515	. . . . . . . . 9 |- ( ( card ` A ) = A -> ( ( card ` A ) e. On <-> A e. On ) )
3	1, 2	mpbii 211	. . . . . . . 8 |- ( ( card ` A ) = A -> A e. On )
4		alephle 8472	. . . . . . . . 9 |- ( A e. On -> A C_ ( aleph ` A ) )
5		fveq2 5856	. . . . . . . . . . 11 |- ( x = A -> ( aleph ` x ) = ( aleph ` A ) )
6	5	sseq2d 3517	. . . . . . . . . 10 |- ( x = A -> ( A C_ ( aleph ` x ) <-> A C_ ( aleph ` A ) ) )
7	6	rspcev 3196	. . . . . . . . 9 |- ( ( A e. On /\ A C_ ( aleph ` A ) ) -> E. x e. On A C_ ( aleph ` x ) )
8	4, 7	mpdan 668	. . . . . . . 8 |- ( A e. On -> E. x e. On A C_ ( aleph ` x ) )
9		nfcv 2605	. . . . . . . . . 10 |- F/_ x A
10		nfcv 2605	. . . . . . . . . . 11 |- F/_ x aleph
11		nfrab1 3024	. . . . . . . . . . . 12 |- F/_ x { x e. On | A C_ ( aleph ` x ) }
12	11	nfint 4281	. . . . . . . . . . 11 |- F/_ x |^| { x e. On | A C_ ( aleph ` x ) }
13	10, 12	nffv 5863	. . . . . . . . . 10 |- F/_ x ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } )
14	9, 13	nfss 3482	. . . . . . . . 9 |- F/ x A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } )
15		fveq2 5856	. . . . . . . . . 10 |- ( x = |^| { x e. On | A C_ ( aleph ` x ) } -> ( aleph ` x ) = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
16	15	sseq2d 3517	. . . . . . . . 9 |- ( x = |^| { x e. On | A C_ ( aleph ` x ) } -> ( A C_ ( aleph ` x ) <-> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
17	14, 16	onminsb 6619	. . . . . . . 8 |- ( E. x e. On A C_ ( aleph ` x ) -> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
18	3, 8, 17	3syl 20	. . . . . . 7 |- ( ( card ` A ) = A -> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
19	18	a1i 11	. . . . . 6 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( ( card ` A ) = A -> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
20		fveq2 5856	. . . . . . . . 9 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = ( aleph ` (/) ) )
21		aleph0 8450	. . . . . . . . 9 |- ( aleph ` (/) ) = om
22	20, 21	syl6eq 2500	. . . . . . . 8 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = om )
23	22	sseq1d 3516	. . . . . . 7 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) C_ A <-> om C_ A ) )
24	23	biimprd 223	. . . . . 6 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( om C_ A -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) C_ A ) )
25	19, 24	anim12d 563	. . . . 5 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( ( ( card ` A ) = A /\ om C_ A ) -> ( A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) /\ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) C_ A ) ) )
26		eqss 3504	. . . . 5 |- ( A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> ( A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) /\ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) C_ A ) )
27	25, 26	syl6ibr 227	. . . 4 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( ( ( card ` A ) = A /\ om C_ A ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
28	27	com12 31	. . 3 |- ( ( ( card ` A ) = A /\ om C_ A ) -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
29	28	ancoms 453	. 2 |- ( ( om C_ A /\ ( card ` A ) = A ) -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
30		vex 3098	. . . . . . . . . . . 12 |- y e. _V
31	30	sucid 4947	. . . . . . . . . . 11 |- y e. suc y
32		eleq2 2516	. . . . . . . . . . 11 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> ( y e. |^| { x e. On | A C_ ( aleph ` x ) } <-> y e. suc y ) )
33	31, 32	mpbiri 233	. . . . . . . . . 10 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> y e. |^| { x e. On | A C_ ( aleph ` x ) } )
34		fveq2 5856	. . . . . . . . . . . 12 |- ( x = y -> ( aleph ` x ) = ( aleph ` y ) )
35	34	sseq2d 3517	. . . . . . . . . . 11 |- ( x = y -> ( A C_ ( aleph ` x ) <-> A C_ ( aleph ` y ) ) )
36	35	onnminsb 6624	. . . . . . . . . 10 |- ( y e. On -> ( y e. |^| { x e. On | A C_ ( aleph ` x ) } -> -. A C_ ( aleph ` y ) ) )
37	33, 36	syl5 32	. . . . . . . . 9 |- ( y e. On -> ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> -. A C_ ( aleph ` y ) ) )
38	37	imp 429	. . . . . . . 8 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> -. A C_ ( aleph ` y ) )
39	38	adantl 466	. . . . . . 7 |- ( ( ( card ` A ) = A /\ ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) ) -> -. A C_ ( aleph ` y ) )
40		fveq2 5856	. . . . . . . . . . 11 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = ( aleph ` suc y ) )
41		alephsuc 8452	. . . . . . . . . . 11 |- ( y e. On -> ( aleph ` suc y ) = (har ` ( aleph ` y ) ) )
42	40, 41	sylan9eqr 2506	. . . . . . . . . 10 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = (har ` ( aleph ` y ) ) )
43	42	eleq2d 2513	. . . . . . . . 9 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> A e. (har ` ( aleph ` y ) ) ) )
44	43	biimpd 207	. . . . . . . 8 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) -> A e. (har ` ( aleph ` y ) ) ) )
45		elharval 7992	. . . . . . . . . 10 |- ( A e. (har ` ( aleph ` y ) ) <-> ( A e. On /\ A ~<_ ( aleph ` y ) ) )
46	45	simprbi 464	. . . . . . . . 9 |- ( A e. (har ` ( aleph ` y ) ) -> A ~<_ ( aleph ` y ) )
47		onenon 8333	. . . . . . . . . . . 12 |- ( A e. On -> A e. dom card )
48	3, 47	syl 16	. . . . . . . . . . 11 |- ( ( card ` A ) = A -> A e. dom card )
49		alephon 8453	. . . . . . . . . . . 12 |- ( aleph ` y ) e. On
50		onenon 8333	. . . . . . . . . . . 12 |- ( ( aleph ` y ) e. On -> ( aleph ` y ) e. dom card )
51	49, 50	ax-mp 5	. . . . . . . . . . 11 |- ( aleph ` y ) e. dom card
52		carddom2 8361	. . . . . . . . . . 11 |- ( ( A e. dom card /\ ( aleph ` y ) e. dom card ) -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A ~<_ ( aleph ` y ) ) )
53	48, 51, 52	sylancl 662	. . . . . . . . . 10 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A ~<_ ( aleph ` y ) ) )
54		sseq1 3510	. . . . . . . . . . 11 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A C_ ( card ` ( aleph ` y ) ) ) )
55		alephcard 8454	. . . . . . . . . . . 12 |- ( card ` ( aleph ` y ) ) = ( aleph ` y )
56	55	sseq2i 3514	. . . . . . . . . . 11 |- ( A C_ ( card ` ( aleph ` y ) ) <-> A C_ ( aleph ` y ) )
57	54, 56	syl6bb 261	. . . . . . . . . 10 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A C_ ( aleph ` y ) ) )
58	53, 57	bitr3d 255	. . . . . . . . 9 |- ( ( card ` A ) = A -> ( A ~<_ ( aleph ` y ) <-> A C_ ( aleph ` y ) ) )
59	46, 58	syl5ib 219	. . . . . . . 8 |- ( ( card ` A ) = A -> ( A e. (har ` ( aleph ` y ) ) -> A C_ ( aleph ` y ) ) )
60	44, 59	sylan9r 658	. . . . . . 7 |- ( ( ( card ` A ) = A /\ ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) -> A C_ ( aleph ` y ) ) )
61	39, 60	mtod 177	. . . . . 6 |- ( ( ( card ` A ) = A /\ ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) ) -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
62	61	rexlimdvaa 2936	. . . . 5 |- ( ( card ` A ) = A -> ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
63		onintrab2 6622	. . . . . . . . . . . . . 14 |- ( E. x e. On A C_ ( aleph ` x ) <-> |^| { x e. On | A C_ ( aleph ` x ) } e. On )
64	8, 63	sylib 196	. . . . . . . . . . . . 13 |- ( A e. On -> |^| { x e. On | A C_ ( aleph ` x ) } e. On )
65		onelon 4893	. . . . . . . . . . . . 13 |- ( ( |^| { x e. On | A C_ ( aleph ` x ) } e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> y e. On )
66	64, 65	sylan 471	. . . . . . . . . . . 12 |- ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> y e. On )
67	36	adantld 467	. . . . . . . . . . . 12 |- ( y e. On -> ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A C_ ( aleph ` y ) ) )
68	66, 67	mpcom 36	. . . . . . . . . . 11 |- ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A C_ ( aleph ` y ) )
69	49	onelssi 4976	. . . . . . . . . . 11 |- ( A e. ( aleph ` y ) -> A C_ ( aleph ` y ) )
70	68, 69	nsyl 121	. . . . . . . . . 10 |- ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A e. ( aleph ` y ) )
71	70	nrexdv 2899	. . . . . . . . 9 |- ( A e. On -> -. E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) )
72	71	adantr 465	. . . . . . . 8 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) )
73		alephlim 8451	. . . . . . . . . . 11 |- ( ( |^| { x e. On | A C_ ( aleph ` x ) } e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = U_ y e. |^| { x e. On | A C_ ( aleph ` x ) } ( aleph ` y ) )
74	64, 73	sylan 471	. . . . . . . . . 10 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = U_ y e. |^| { x e. On | A C_ ( aleph ` x ) } ( aleph ` y ) )
75	74	eleq2d 2513	. . . . . . . . 9 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> A e. U_ y e. |^| { x e. On | A C_ ( aleph ` x ) } ( aleph ` y ) ) )
76		eliun 4320	. . . . . . . . 9 |- ( A e. U_ y e. |^| { x e. On | A C_ ( aleph ` x ) } ( aleph ` y ) <-> E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) )
77	75, 76	syl6bb 261	. . . . . . . 8 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) ) )
78	72, 77	mtbird 301	. . . . . . 7 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
79	78	ex 434	. . . . . 6 |- ( A e. On -> ( Lim |^| { x e. On | A C_ ( aleph ` x ) } -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
80	3, 79	syl 16	. . . . 5 |- ( ( card ` A ) = A -> ( Lim |^| { x e. On | A C_ ( aleph ` x ) } -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
81	62, 80	jaod 380	. . . 4 |- ( ( card ` A ) = A -> ( ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
82	8, 17	syl 16	. . . . . 6 |- ( A e. On -> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
83		alephon 8453	. . . . . . 7 |- ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) e. On
84		onsseleq 4909	. . . . . . 7 |- ( ( A e. On /\ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) e. On ) -> ( A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) \/ A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) ) )
85	83, 84	mpan2 671	. . . . . 6 |- ( A e. On -> ( A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) \/ A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) ) )
86	82, 85	mpbid 210	. . . . 5 |- ( A e. On -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) \/ A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
87	86	ord 377	. . . 4 |- ( A e. On -> ( -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
88	3, 81, 87	sylsyld 56	. . 3 |- ( ( card ` A ) = A -> ( ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
89	88	adantl 466	. 2 |- ( ( om C_ A /\ ( card ` A ) = A ) -> ( ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
90		eloni 4878	. . . . 5 |- ( |^| { x e. On | A C_ ( aleph ` x ) } e. On -> Ord |^| { x e. On | A C_ ( aleph ` x ) } )
91		ordzsl 6665	. . . . . 6 |- ( Ord |^| { x e. On | A C_ ( aleph ` x ) } <-> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) )
92		3orass 977	. . . . . 6 |- ( ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) <-> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
93	91, 92	bitri 249	. . . . 5 |- ( Ord |^| { x e. On | A C_ ( aleph ` x ) } <-> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
94	90, 93	sylib 196	. . . 4 |- ( |^| { x e. On | A C_ ( aleph ` x ) } e. On -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
95	3, 64, 94	3syl 20	. . 3 |- ( ( card ` A ) = A -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
96	95	adantl 466	. 2 |- ( ( om C_ A /\ ( card ` A ) = A ) -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
97	29, 89, 96	mpjaod 381	1 |- ( ( om C_ A /\ ( card ` A ) = A ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/ w3o 973   = wceq 1383   e. wcel 1804   E.wrex 2794   {crab 2797   C_ wss 3461   (/)c0 3770   |^|cint 4271   U_ciun 4315   class class class wbr 4437   Ord word 4867   Oncon0 4868   Lim wlim 4869   suc csuc 4870   dom cdm 4989   ` cfv 5578   omcom 6685   ~<_ cdom 7516  harchar 7985   cardccrd 8319   alephcale 8320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-har 7987  df-card 8323  df-aleph 8324

This theorem is referenced by:  cardalephex  8474  tskcard  9162