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Theorem cantnflt2OLD 8139
 Description: An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnflt2 8109 as of 29-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1 CNF
cantnfsOLD.2
cantnfsOLD.3
cantnflt2OLD.4
cantnflt2OLD.5
cantnflt2OLD.6
cantnflt2OLD.7
Assertion
Ref Expression
cantnflt2OLD CNF

Proof of Theorem cantnflt2OLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfsOLD.1 . . 3 CNF
2 cantnfsOLD.2 . . 3
3 cantnfsOLD.3 . . 3
4 eqid 2457 . . 3 OrdIso OrdIso
5 cantnflt2OLD.4 . . 3
6 eqid 2457 . . 3 seq𝜔 OrdIso OrdIso seq𝜔 OrdIso OrdIso
71, 2, 3, 4, 5, 6cantnfvalOLD 8134 . 2 CNF seq𝜔 OrdIso OrdIso OrdIso
8 cantnflt2OLD.5 . . 3
9 cantnflt2OLD.6 . . . . 5
10 cantnflt2OLD.7 . . . . 5
119, 10ssexd 4603 . . . 4
124oion 7979 . . . 4 OrdIso
13 sucidg 4965 . . . 4 OrdIso OrdIso OrdIso
1411, 12, 133syl 20 . . 3 OrdIso OrdIso
151, 2, 3, 4, 5cantnfclOLD 8133 . . . . . . 7 OrdIso
1615simpld 459 . . . . . 6
174oiiso 7980 . . . . . 6 OrdIso OrdIso
1811, 16, 17syl2anc 661 . . . . 5 OrdIso OrdIso
19 isof1o 6222 . . . . 5 OrdIso OrdIso OrdIso OrdIso
20 f1ofo 5829 . . . . 5 OrdIso OrdIso OrdIso OrdIso
21 foima 5806 . . . . 5 OrdIso OrdIso OrdIso OrdIso
2218, 19, 20, 214syl 21 . . . 4 OrdIso OrdIso
2322, 10eqsstrd 3533 . . 3 OrdIso OrdIso
241, 2, 3, 4, 5, 6, 8, 14, 9, 23cantnfltOLD 8138 . 2 seq𝜔 OrdIso OrdIso OrdIso
257, 24eqeltrd 2545 1 CNF
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1395   wcel 1819  cvv 3109   cdif 3468   wss 3471  c0 3793   cep 4798   wwe 4846  con0 4887   csuc 4889  ccnv 5007   cdm 5008  cima 5011  wfo 5592  wf1o 5593  cfv 5594   wiso 5595  (class class class)co 6296   cmpt2 6298  com 6699  seq𝜔cseqom 7130  c1o 7141   coa 7145   comu 7146   coe 7147  OrdIsocoi 7952   CNF ccnf 8095 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 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 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-oexp 7154  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-cnf 8096 This theorem is referenced by:  cantnflem1dOLD  8147  cnfcom3lemOLD  8172
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