Theorem infxpen 8413

Description: Every infinite ordinal is equinumerous to its Cartesian product. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation is a well-ordering of with the additional property that -initial segments of (where is a limit ordinal) are of cardinality at most . (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
infxpen

Proof of Theorem infxpen

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

Step	Hyp	Ref	Expression
1		eqid 2457	. 2
2		eleq1 2529	. . . . 5
3		eleq1 2529	. . . . 5
4	2, 3	bi2anan9 873	. . . 4
5		fveq2 5871	. . . . . . . 8
6		fveq2 5871	. . . . . . . 8
7	5, 6	uneq12d 3658	. . . . . . 7
8	7	adantr 465	. . . . . 6
9		fveq2 5871	. . . . . . . 8
10		fveq2 5871	. . . . . . . 8
11	9, 10	uneq12d 3658	. . . . . . 7
12	11	adantl 466	. . . . . 6
13	8, 12	eleq12d 2539	. . . . 5
14	7, 11	eqeqan12d 2480	. . . . . 6
15		breq12 4457	. . . . . 6
16	14, 15	anbi12d 710	. . . . 5
17	13, 16	orbi12d 709	. . . 4
18	4, 17	anbi12d 710	. . 3
19	18	cbvopabv 4521	. 2
20		eqid 2457	. 2
21		biid 236	. 2
22		eqid 2457	. 2
23		eqid 2457	. 2
24	1, 19, 20, 21, 22, 23	infxpenlem 8412	1

Syntax hints:  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  u.cun 3473  i^icin 3474  C_wss 3475   class class class wbr 4452  {copab 4509  con0 4883  X.cxp 5002  `cfv 5593  com 6700  c1st 6798  c2nd 6799  cen 7533  csdm 7535  OrdIsocoi 7955

This theorem is referenced by:  xpomen  8414  infxpidm2  8415  alephreg  8978  cfpwsdom  8980  inar1  9174

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-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-inf2 8079

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-rmo 2815  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-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-se 4844  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-isom 5602  df-riota 6257  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6701  df-1st 6800  df-2nd 6801  df-recs 7061  df-rdg 7095  df-1o 7149  df-oadd 7153  df-er 7330  df-en 7537  df-dom 7538  df-sdom 7539  df-fin 7540  df-oi 7956  df-card 8341