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Theorem axdc3lem 8829
 Description: The class of finite approximations to the DC sequence is a set. (We derive here the stronger statement that is a subset of a specific set, namely .) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 18-Mar-2014.)
Hypotheses
Ref Expression
axdc3lem.1
axdc3lem.2
Assertion
Ref Expression
axdc3lem
Distinct variable group:   ,,
Allowed substitution hints:   ()   (,,)   (,,)   (,,)

Proof of Theorem axdc3lem
StepHypRef Expression
1 dcomex 8826 . . . 4
2 axdc3lem.1 . . . 4
31, 2xpex 6712 . . 3
43pwex 4630 . 2
5 axdc3lem.2 . . 3
6 fssxp 5742 . . . . . . . . 9
7 peano2 6699 . . . . . . . . . 10
8 omelon2 6691 . . . . . . . . . . . 12
91, 8ax-mp 5 . . . . . . . . . . 11
109onelssi 4986 . . . . . . . . . 10
11 xpss1 5110 . . . . . . . . . 10
127, 10, 113syl 20 . . . . . . . . 9
136, 12sylan9ss 3517 . . . . . . . 8
14 selpw 4017 . . . . . . . 8
1513, 14sylibr 212 . . . . . . 7
1615ancoms 453 . . . . . 6
17163ad2antr1 1161 . . . . 5
1817rexlimiva 2951 . . . 4
1918abssi 3575 . . 3
205, 19eqsstri 3534 . 2
214, 20ssexi 4592 1
 Colors of variables: wff setvar class Syntax hints:   wa 369   w3a 973   wceq 1379   wcel 1767  cab 2452  wral 2814  wrex 2815  cvv 3113   wss 3476  c0 3785  cpw 4010  con0 4878   csuc 4880   cxp 4997  wf 5583  cfv 5587  com 6679 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-dc 8825 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-fv 5595  df-om 6680  df-1o 7130 This theorem is referenced by:  axdc3lem2  8830  axdc3lem4  8832
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