Theorem karden 8334

Description: If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 8947). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 8333 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from . (Contributed by NM, 18-Dec-2003.)

Hypotheses

Ref	Expression
karden.1
karden.2
karden.3
karden.4

Assertion

Ref	Expression
karden

Distinct variable groups:   ,,   ,,

Proof of Theorem karden

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		karden.1	. . . . . . . 8
2	1	enref 7568	. . . . . . 7
3		breq1 4455	. . . . . . . 8
4	1, 3	spcev 3201	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6		abn0 3804	. . . . . 6
7	5, 6	mpbir 209	. . . . 5
8		scott0 8325	. . . . . 6
9	8	necon3bii 2725	. . . . 5
10	7, 9	mpbi 208	. . . 4
11		rabn0 3805	. . . 4
12	10, 11	mpbi 208	. . 3
13		vex 3112	. . . . . . . 8
14		breq1 4455	. . . . . . . 8
15	13, 14	elab 3246	. . . . . . 7
16		breq1 4455	. . . . . . . 8
17	16	ralab 3260	. . . . . . 7
18	15, 17	anbi12i 697	. . . . . 6
19		simpl 457	. . . . . . . . 9
20	19	a1i 11	. . . . . . . 8
21		karden.3	. . . . . . . . . . . 12
22		karden.4	. . . . . . . . . . . 12
23	21, 22	eqeq12i 2477	. . . . . . . . . . 11
24		abbi 2588	. . . . . . . . . . 11
25	23, 24	bitr4i 252	. . . . . . . . . 10
26		breq1 4455	. . . . . . . . . . . . 13
27		fveq2 5871	. . . . . . . . . . . . . . . 16
28	27	sseq1d 3530	. . . . . . . . . . . . . . 15
29	28	imbi2d 316	. . . . . . . . . . . . . 14
30	29	albidv 1713	. . . . . . . . . . . . 13
31	26, 30	anbi12d 710	. . . . . . . . . . . 12
32		breq1 4455	. . . . . . . . . . . . 13
33	28	imbi2d 316	. . . . . . . . . . . . . 14
34	33	albidv 1713	. . . . . . . . . . . . 13
35	32, 34	anbi12d 710	. . . . . . . . . . . 12
36	31, 35	bibi12d 321	. . . . . . . . . . 11
37	36	spv 2011	. . . . . . . . . 10
38	25, 37	sylbi 195	. . . . . . . . 9
39		simpl 457	. . . . . . . . 9
40	38, 39	syl6bi 228	. . . . . . . 8
41	20, 40	jcad 533	. . . . . . 7
42		ensym 7584	. . . . . . . 8
43		entr 7587	. . . . . . . 8
44	42, 43	sylan 471	. . . . . . 7
45	41, 44	syl6 33	. . . . . 6
46	18, 45	syl5bi 217	. . . . 5
47	46	expd 436	. . . 4
48	47	rexlimdv 2947	. . 3
49	12, 48	mpi 17	. 2
50		enen2 7678	. . . . 5
51		enen2 7678	. . . . . . 7
52	51	imbi1d 317	. . . . . 6
53	52	albidv 1713	. . . . 5
54	50, 53	anbi12d 710	. . . 4
55	54	abbidv 2593	. . 3
56	55, 21, 22	3eqtr4g 2523	. 2
57	49, 56	impbii 188	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  =/=wne 2652  A.wral 2807  E.wrex 2808  {crab 2811  cvv 3109  C_wss 3475  c0 3784   class class class wbr 4452  `cfv 5593  cen 7533  crnk 8202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-om 6701  df-recs 7061  df-rdg 7095  df-er 7330  df-en 7537  df-r1 8203  df-rank 8204