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Theorem axdc3lem 8878
 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 8875 . . . 4
2 axdc3lem.1 . . . 4
31, 2xpex 6609 . . 3
43pwex 4608 . 2
5 axdc3lem.2 . . 3
6 fssxp 5758 . . . . . . . . 9
7 peano2 6727 . . . . . . . . . 10
8 omelon2 6718 . . . . . . . . . . . 12
91, 8ax-mp 5 . . . . . . . . . . 11
109onelssi 5550 . . . . . . . . . 10
11 xpss1 4963 . . . . . . . . . 10
127, 10, 113syl 18 . . . . . . . . 9
136, 12sylan9ss 3483 . . . . . . . 8
14 selpw 3992 . . . . . . . 8
1513, 14sylibr 215 . . . . . . 7
1615ancoms 454 . . . . . 6
17163ad2antr1 1170 . . . . 5
1817rexlimiva 2920 . . . 4
1918abssi 3542 . . 3
205, 19eqsstri 3500 . 2
214, 20ssexi 4570 1
 Colors of variables: wff setvar class Syntax hints:   wa 370   w3a 982   wceq 1437   wcel 1870  cab 2414  wral 2782  wrex 2783  cvv 3087   wss 3442  c0 3767  cpw 3985   cxp 4852  con0 5442   csuc 5444  wf 5597  cfv 5601  com 6706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-dc 8874 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-om 6707  df-1o 7190 This theorem is referenced by:  axdc3lem2  8879  axdc3lem4  8881
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