Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnfclOLD Structured version   Unicode version

Theorem cantnfclOLD 8068
 Description: Basic properties of the order isomorphism used later. The support of an is a finite subset of , so it is well-ordered by and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnfcl 8038 as of 28-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1 CNF
cantnfsOLD.2
cantnfsOLD.3
cantnfvalOLD.3 OrdIso
cantnfvalOLD.4
Assertion
Ref Expression
cantnfclOLD

Proof of Theorem cantnfclOLD
StepHypRef Expression
1 cnvimass 5298 . . . . 5
2 cantnfvalOLD.4 . . . . . . . 8
3 cantnfsOLD.1 . . . . . . . . 9 CNF
4 cantnfsOLD.2 . . . . . . . . 9
5 cantnfsOLD.3 . . . . . . . . 9
63, 4, 5cantnfsOLD 8067 . . . . . . . 8
72, 6mpbid 210 . . . . . . 7
87simpld 457 . . . . . 6
9 fdm 5674 . . . . . 6
108, 9syl 17 . . . . 5
111, 10syl5sseq 3489 . . . 4
12 onss 6564 . . . . 5
135, 12syl 17 . . . 4
1411, 13sstrd 3451 . . 3
15 epweon 6557 . . 3
16 wess 4809 . . 3
1714, 15, 16mpisyl 19 . 2
18 cnvexg 6684 . . . . 5
19 imaexg 6675 . . . . 5
20 cantnfvalOLD.3 . . . . . 6 OrdIso
2120oion 7915 . . . . 5
222, 18, 19, 214syl 21 . . . 4
237simprd 461 . . . . 5
2420oien 7917 . . . . . 6
2523, 17, 24syl2anc 659 . . . . 5
26 enfii 7692 . . . . 5
2723, 25, 26syl2anc 659 . . . 4
2822, 27elind 3626 . . 3
29 onfin2 7667 . . 3
3028, 29syl6eleqr 2501 . 2
3117, 30jca 530 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   wceq 1405   wcel 1842  cvv 3058   cdif 3410   cin 3412   wss 3413   class class class wbr 4394   cep 4731   wwe 4780  con0 4821  ccnv 4941   cdm 4942  cima 4945  wf 5521  (class class class)co 6234  com 6638  c1o 7080   cen 7471  cfn 7474  OrdIsocoi 7888   CNF ccnf 8030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-supp 6857  df-recs 6999  df-rdg 7033  df-seqom 7070  df-1o 7087  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-fsupp 7784  df-oi 7889  df-cnf 8031 This theorem is referenced by:  cantnfval2OLD  8070  cantnfleOLD  8072  cantnfltOLD  8073  cantnflt2OLD  8074  cantnfp1lem2OLD  8076  cantnfp1lem3OLD  8077  cantnflem1bOLD  8080  cantnflem1dOLD  8082  cantnflem1OLD  8083  cnfcomlemOLD  8103  cnfcomOLD  8104  cnfcom2lemOLD  8105  cnfcom3lemOLD  8107
 Copyright terms: Public domain W3C validator