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Theorem infxpencOLD 8403
 Description: A canonical version of infxpen 8395, by a completely different approach (although it uses infxpen 8395 via xpomen 8396). Using Cantor's normal form, we can show that respects equinumerosity (oef1oOLD 8145), so that all the steps of can be verified using bijections to do the ordinal commutations. (The assumption on can be satisfied using cnfcom3cOLD 8161.) (Contributed by Mario Carneiro, 30-May-2015.) Obsolete version of infxpenc 8398 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 CNF CNF
infxpencOLD.l
infxpencOLD.x
infxpencOLD.y
infxpencOLD.j CNF CNF
infxpencOLD.z
infxpencOLD.t
infxpencOLD.g
Assertion
Ref Expression
infxpencOLD
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,,,)   (,)   (,,,)   (,,,)   (,,,)   (,,,)   (,,,)   (,)   (,)   (,)   (,,,)

Proof of Theorem infxpencOLD
StepHypRef Expression
1 infxpencOLD.6 . . . 4
2 f1ocnv 5818 . . . 4
31, 2syl 16 . . 3
4 infxpencOLD.4 . . . . . . . 8
5 f1oi 5841 . . . . . . . . 9
65a1i 11 . . . . . . . 8
7 omelon 8066 . . . . . . . . . . 11
87a1i 11 . . . . . . . . . 10
9 2on 7140 . . . . . . . . . 10
10 oecl 7189 . . . . . . . . . 10
118, 9, 10sylancl 662 . . . . . . . . 9
129a1i 11 . . . . . . . . . 10
13 peano1 6704 . . . . . . . . . . 11
1413a1i 11 . . . . . . . . . 10
15 oen0 7237 . . . . . . . . . 10
168, 12, 14, 15syl21anc 1228 . . . . . . . . 9
17 ondif1 7153 . . . . . . . . 9
1811, 16, 17sylanbrc 664 . . . . . . . 8
19 infxpencOLD.3 . . . . . . . . 9
2019eldifad 3473 . . . . . . . 8
21 infxpencOLD.5 . . . . . . . 8
22 infxpencOLD.k . . . . . . . 8
23 infxpencOLD.h . . . . . . . 8 CNF CNF
244, 6, 18, 20, 8, 20, 21, 22, 23oef1oOLD 8145 . . . . . . 7
25 f1oi 5841 . . . . . . . . . 10
2625a1i 11 . . . . . . . . 9
27 infxpencOLD.x . . . . . . . . . . 11
28 infxpencOLD.y . . . . . . . . . . 11
2927, 28omf1o 7622 . . . . . . . . . 10
3020, 9, 29sylancl 662 . . . . . . . . 9
31 ondif1 7153 . . . . . . . . . . 11
327, 13, 31mpbir2an 920 . . . . . . . . . 10
3332a1i 11 . . . . . . . . 9
34 omcl 7188 . . . . . . . . . 10
3520, 9, 34sylancl 662 . . . . . . . . 9
36 omcl 7188 . . . . . . . . . 10
3712, 20, 36syl2anc 661 . . . . . . . . 9
38 fvresi 6082 . . . . . . . . . 10
3913, 38mp1i 12 . . . . . . . . 9
40 infxpencOLD.l . . . . . . . . 9
41 infxpencOLD.j . . . . . . . . 9 CNF CNF
4226, 30, 33, 35, 8, 37, 39, 40, 41oef1oOLD 8145 . . . . . . . 8
43 oeoe 7250 . . . . . . . . . 10
448, 12, 20, 43syl3anc 1229 . . . . . . . . 9
45 f1oeq3 5799 . . . . . . . . 9
4644, 45syl 16 . . . . . . . 8
4742, 46mpbird 232 . . . . . . 7
48 f1oco 5828 . . . . . . 7
4924, 47, 48syl2anc 661 . . . . . 6
50 df-2o 7133 . . . . . . . . . . . 12
5150oveq2i 6292 . . . . . . . . . . 11
52 1on 7139 . . . . . . . . . . . 12
53 omsuc 7178 . . . . . . . . . . . 12
5420, 52, 53sylancl 662 . . . . . . . . . . 11
5551, 54syl5eq 2496 . . . . . . . . . 10
56 om1 7193 . . . . . . . . . . . 12
5720, 56syl 16 . . . . . . . . . . 11
5857oveq1d 6296 . . . . . . . . . 10
5955, 58eqtrd 2484 . . . . . . . . 9
6059oveq2d 6297 . . . . . . . 8
61 oeoa 7248 . . . . . . . . 9
628, 20, 20, 61syl3anc 1229 . . . . . . . 8
6360, 62eqtrd 2484 . . . . . . 7
64 f1oeq2 5798 . . . . . . 7
6563, 64syl 16 . . . . . 6
6649, 65mpbid 210 . . . . 5
67 oecl 7189 . . . . . . 7
688, 20, 67syl2anc 661 . . . . . 6
69 infxpencOLD.z . . . . . . 7
7069omxpenlem 7620 . . . . . 6
7168, 68, 70syl2anc 661 . . . . 5
72 f1oco 5828 . . . . 5
7366, 71, 72syl2anc 661 . . . 4
74 f1of 5806 . . . . . . . . . 10
751, 74syl 16 . . . . . . . . 9
7675feqmptd 5911 . . . . . . . 8
77 f1oeq1 5797 . . . . . . . 8
7876, 77syl 16 . . . . . . 7
791, 78mpbid 210 . . . . . 6
8075feqmptd 5911 . . . . . . . 8
81 f1oeq1 5797 . . . . . . . 8
8280, 81syl 16 . . . . . . 7
831, 82mpbid 210 . . . . . 6
8479, 83xpf1o 7681 . . . . 5
85 infxpencOLD.t . . . . . 6
86 f1oeq1 5797 . . . . . 6
8785, 86ax-mp 5 . . . . 5
8884, 87sylibr 212 . . . 4
89 f1oco 5828 . . . 4
9073, 88, 89syl2anc 661 . . 3
91 f1oco 5828 . . 3
923, 90, 91syl2anc 661 . 2
93 infxpencOLD.g . . 3
94 f1oeq1 5797 . . 3
9593, 94ax-mp 5 . 2
9692, 95sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wceq 1383   wcel 1804  crab 2797  cvv 3095   cdif 3458   wss 3461  c0 3770  cop 4020   cmpt 4495   cid 4780  con0 4868   csuc 4870   cxp 4987  ccnv 4988   cres 4991  cima 4992   ccom 4993  wf 5574  wf1o 5577  cfv 5578  (class class class)co 6281   cmpt2 6283  com 6685  c1o 7125  c2o 7126   coa 7129   comu 7130   coe 7131   cmap 7422  cfn 7518   CNF ccnf 8081 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-fal 1389  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-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-seqom 7115  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-cnf 8082 This theorem is referenced by:  infxpenc2lem2OLD  8404
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