Theorem infxpencOLD 8421

Description: A canonical version of infxpen 8413, by a completely different approach (although it uses infxpen 8413 via xpomen 8414). Using Cantor's normal form, we can show that respects equinumerosity (oef1oOLD 8163), so that all the steps of ()()(2) (2) can be verified using bijections to do the ordinal commutations. (The assumption on can be satisfied using cnfcom3cOLD 8179.) (Contributed by Mario Carneiro, 30-May-2015.) Obsolete version of infxpenc 8416 as of 7-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
infxpencOLD.1
infxpencOLD.2
infxpencOLD.3
infxpencOLD.4
infxpencOLD.5
infxpencOLD.6
infxpencOLD.k
infxpencOLD.h
infxpencOLD.l
infxpencOLD.x
infxpencOLD.y
infxpencOLD.j
infxpencOLD.z
infxpencOLD.t
infxpencOLD.g

Assertion

Ref	Expression
infxpencOLD

Distinct variable groups:   ,,   ,,   ,N,   ,,   ,,,,   ,,   ,,

Proof of Theorem infxpencOLD

Step	Hyp	Ref	Expression
1		infxpencOLD.6	. . . 4
2		f1ocnv 5833	. . . 4
3	1, 2	syl 16	. . 3
4		infxpencOLD.4	. . . . . . . 8
5		f1oi 5856	. . . . . . . . 9
6	5	a1i 11	. . . . . . . 8
7		omelon 8084	. . . . . . . . . . 11
8	7	a1i 11	. . . . . . . . . 10
9		2on 7157	. . . . . . . . . 10
10		oecl 7206	. . . . . . . . . 10
11	8, 9, 10	sylancl 662	. . . . . . . . 9
12	9	a1i 11	. . . . . . . . . 10
13		peano1 6719	. . . . . . . . . . 11
14	13	a1i 11	. . . . . . . . . 10
15		oen0 7254	. . . . . . . . . 10
16	8, 12, 14, 15	syl21anc 1227	. . . . . . . . 9
17		ondif1 7170	. . . . . . . . 9
18	11, 16, 17	sylanbrc 664	. . . . . . . 8
19		infxpencOLD.3	. . . . . . . . 9
20	19	eldifad 3487	. . . . . . . 8
21		infxpencOLD.5	. . . . . . . 8
22		infxpencOLD.k	. . . . . . . 8
23		infxpencOLD.h	. . . . . . . 8
24	4, 6, 18, 20, 8, 20, 21, 22, 23	oef1oOLD 8163	. . . . . . 7
25		f1oi 5856	. . . . . . . . . 10
26	25	a1i 11	. . . . . . . . 9
27		infxpencOLD.x	. . . . . . . . . . 11
28		infxpencOLD.y	. . . . . . . . . . 11
29	27, 28	omf1o 7640	. . . . . . . . . 10
30	20, 9, 29	sylancl 662	. . . . . . . . 9
31		ondif1 7170	. . . . . . . . . . 11
32	7, 13, 31	mpbir2an 920	. . . . . . . . . 10
33	32	a1i 11	. . . . . . . . 9
34		omcl 7205	. . . . . . . . . 10
35	20, 9, 34	sylancl 662	. . . . . . . . 9
36		omcl 7205	. . . . . . . . . 10
37	12, 20, 36	syl2anc 661	. . . . . . . . 9
38		fvresi 6097	. . . . . . . . . 10
39	13, 38	mp1i 12	. . . . . . . . 9
40		infxpencOLD.l	. . . . . . . . 9
41		infxpencOLD.j	. . . . . . . . 9
42	26, 30, 33, 35, 8, 37, 39, 40, 41	oef1oOLD 8163	. . . . . . . 8
43		oeoe 7267	. . . . . . . . . 10
44	8, 12, 20, 43	syl3anc 1228	. . . . . . . . 9
45		f1oeq3 5814	. . . . . . . . 9
46	44, 45	syl 16	. . . . . . . 8
47	42, 46	mpbird 232	. . . . . . 7
48		f1oco 5843	. . . . . . 7
49	24, 47, 48	syl2anc 661	. . . . . 6
50		df-2o 7150	. . . . . . . . . . . 12
51	50	oveq2i 6307	. . . . . . . . . . 11
52		1on 7156	. . . . . . . . . . . 12
53		omsuc 7195	. . . . . . . . . . . 12
54	20, 52, 53	sylancl 662	. . . . . . . . . . 11
55	51, 54	syl5eq 2510	. . . . . . . . . 10
56		om1 7210	. . . . . . . . . . . 12
57	20, 56	syl 16	. . . . . . . . . . 11
58	57	oveq1d 6311	. . . . . . . . . 10
59	55, 58	eqtrd 2498	. . . . . . . . 9
60	59	oveq2d 6312	. . . . . . . 8
61		oeoa 7265	. . . . . . . . 9
62	8, 20, 20, 61	syl3anc 1228	. . . . . . . 8
63	60, 62	eqtrd 2498	. . . . . . 7
64		f1oeq2 5813	. . . . . . 7
65	63, 64	syl 16	. . . . . 6
66	49, 65	mpbid 210	. . . . 5
67		oecl 7206	. . . . . . 7
68	8, 20, 67	syl2anc 661	. . . . . 6
69		infxpencOLD.z	. . . . . . 7
70	69	omxpenlem 7638	. . . . . 6
71	68, 68, 70	syl2anc 661	. . . . 5
72		f1oco 5843	. . . . 5
73	66, 71, 72	syl2anc 661	. . . 4
74		f1of 5821	. . . . . . . . . 10
75	1, 74	syl 16	. . . . . . . . 9
76	75	feqmptd 5926	. . . . . . . 8
77		f1oeq1 5812	. . . . . . . 8
78	76, 77	syl 16	. . . . . . 7
79	1, 78	mpbid 210	. . . . . 6
80	75	feqmptd 5926	. . . . . . . 8
81		f1oeq1 5812	. . . . . . . 8
82	80, 81	syl 16	. . . . . . 7
83	1, 82	mpbid 210	. . . . . 6
84	79, 83	xpf1o 7699	. . . . 5
85		infxpencOLD.t	. . . . . 6
86		f1oeq1 5812	. . . . . 6
87	85, 86	ax-mp 5	. . . . 5
88	84, 87	sylibr 212	. . . 4
89		f1oco 5843	. . . 4
90	73, 88, 89	syl2anc 661	. . 3
91		f1oco 5843	. . 3
92	3, 90, 91	syl2anc 661	. 2
93		infxpencOLD.g	. . 3
94		f1oeq1 5812	. . 3
95	93, 94	ax-mp 5	. 2
96	92, 95	sylibr 212	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  {crab 2811  cvv 3109  \cdif 3472  C_wss 3475  c0 3784  <.cop 4035  e.cmpt 4510  cid 4795  con0 4883  succsuc 4885  X.cxp 5002  `'ccnv 5003  |`cres 5006  "cima 5007  o.ccom 5008  -->wf 5589  -1-1-onto->wf1o 5592  `cfv 5593  (class class class)co 6296  e.cmpt2 6298  com 6700  c1o 7142  c2o 7143  coa 7146  comu 7147  coe 7148  cmap 7439  cfn 7536  ccnf 8099

This theorem is referenced by:  infxpenc2lem2OLD  8422

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-fal 1401  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-supp 6919  df-recs 7061  df-rdg 7095  df-seqom 7132  df-1o 7149  df-2o 7150  df-oadd 7153  df-omul 7154  df-oexp 7155  df-er 7330  df-map 7441  df-en 7537  df-dom 7538  df-sdom 7539  df-fin 7540  df-fsupp 7850  df-oi 7956  df-cnf 8100