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Theorem cantnffval2OLD 8039
 Description: An alternative definition of df-cnf 7982 which relies on cantnfOLD 8037. (Note that although the use of seems self-referential, one can use cantnfdmOLD 7986 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnffval2 8017 as of 2-Jul-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1 CNF
cantnfsOLD.2
cantnfsOLD.3
oemapvalOLD.t
Assertion
Ref Expression
cantnffval2OLD CNF OrdIso
Distinct variable groups:   ,,,,   ,,,,   ,,,   ,,,
Allowed substitution hints:   ()   ()   (,,,)

Proof of Theorem cantnffval2OLD
StepHypRef Expression
1 cantnfsOLD.1 . . . . 5 CNF
2 cantnfsOLD.2 . . . . 5
3 cantnfsOLD.3 . . . . 5
4 oemapvalOLD.t . . . . 5
51, 2, 3, 4cantnfOLD 8037 . . . 4 CNF
6 isof1o 6128 . . . 4 CNF CNF
7 f1orel 5755 . . . 4 CNF CNF
85, 6, 73syl 20 . . 3 CNF
9 dfrel2 5399 . . 3 CNF CNF CNF
108, 9sylib 196 . 2 CNF CNF
11 oecl 7090 . . . . . . 7
122, 3, 11syl2anc 661 . . . . . 6
13 eloni 4840 . . . . . 6
1412, 13syl 16 . . . . 5
15 isocnv 6133 . . . . . 6 CNF CNF
165, 15syl 16 . . . . 5 CNF
171, 2, 3, 4oemapwe 8016 . . . . . . 7 OrdIso
1817simpld 459 . . . . . 6
19 ovex 6228 . . . . . . . . 9 CNF
2019dmex 6624 . . . . . . . 8 CNF
211, 20eqeltri 2538 . . . . . . 7
22 exse 4795 . . . . . . 7 Se
2321, 22ax-mp 5 . . . . . 6 Se
24 eqid 2454 . . . . . . 7 OrdIso OrdIso
2524oieu 7867 . . . . . 6 Se CNF OrdIso CNF OrdIso
2618, 23, 25sylancl 662 . . . . 5 CNF OrdIso CNF OrdIso
2714, 16, 26mpbi2and 912 . . . 4 OrdIso CNF OrdIso
2827simprd 463 . . 3 CNF OrdIso
2928cnveqd 5126 . 2 CNF OrdIso
3010, 29eqtr3d 2497 1 CNF OrdIso
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1370   wcel 1758  wral 2799  wrex 2800  cvv 3078  copab 4460   cep 4741   Se wse 4788   wwe 4789   word 4829  con0 4830  ccnv 4950   cdm 4951   wrel 4956  wf1o 5528  cfv 5529   wiso 5530  (class class class)co 6203   coe 7032  OrdIsocoi 7837   CNF ccnf 7981 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-oexp 7039  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7838  df-cnf 7982 This theorem is referenced by: (None)
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