Theorem axdc3lem 8851

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:   ,,

Proof of Theorem axdc3lem

Step	Hyp	Ref	Expression
1		dcomex 8848	. . . 4
2		axdc3lem.1	. . . 4
3	1, 2	xpex 6604	. . 3
4	3	pwex 4635	. 2
5		axdc3lem.2	. . 3
6		fssxp 5748	. . . . . . . . 9
7		peano2 6720	. . . . . . . . . 10
8		omelon2 6712	. . . . . . . . . . . 12
9	1, 8	ax-mp 5	. . . . . . . . . . 11
10	9	onelssi 4991	. . . . . . . . . 10
11		xpss1 5116	. . . . . . . . . 10
12	7, 10, 11	3syl 20	. . . . . . . . 9
13	6, 12	sylan9ss 3516	. . . . . . . 8
14		selpw 4019	. . . . . . . 8
15	13, 14	sylibr 212	. . . . . . 7
16	15	ancoms 453	. . . . . 6
17	16	3ad2antr1 1161	. . . . 5
18	17	rexlimiva 2945	. . . 4
19	18	abssi 3574	. . 3
20	5, 19	eqsstri 3533	. 2
21	4, 20	ssexi 4597	1

Syntax hints:  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808  cvv 3109  C_wss 3475  c0 3784  ~Pcpw 4012  con0 4883  succsuc 4885  X.cxp 5002  -->wf 5589  `cfv 5593  com 6700

This theorem is referenced by:  axdc3lem2  8852  axdc3lem4  8854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-dc 8847

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-om 6701  df-1o 7149