Theorem prcdnq 9392

Description: A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
prcdnq

Proof of Theorem prcdnq

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 9325	. . . . . . 7
2		relxp 5115	. . . . . . 7
3		relss 5095	. . . . . . 7
4	1, 2, 3	mp2 9	. . . . . 6
5	4	brrelexi 5045	. . . . 5
6		eleq1 2529	. . . . . . . . 9
7	6	anbi2d 703	. . . . . . . 8
8		breq2 4456	. . . . . . . 8
9	7, 8	anbi12d 710	. . . . . . 7
10	9	imbi1d 317	. . . . . 6
11		breq1 4455	. . . . . . . 8
12	11	anbi2d 703	. . . . . . 7
13		eleq1 2529	. . . . . . 7
14	12, 13	imbi12d 320	. . . . . 6
15		elnpi 9387	. . . . . . . . . . 11
16	15	simprbi 464	. . . . . . . . . 10
17	16	r19.21bi 2826	. . . . . . . . 9
18	17	simpld 459	. . . . . . . 8
19	18	19.21bi 1869	. . . . . . 7
20	19	imp 429	. . . . . 6
21	10, 14, 20	vtocl2g 3171	. . . . 5
22	5, 21	sylan2 474	. . . 4
23	22	adantll 713	. . 3
24	23	pm2.43i 47	. 2
25	24	ex 434	1

Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  A.wal 1393  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109  C_wss 3475  C.wpss 3476  c0 3784   class class class wbr 4452  X.cxp 5002  Relwrel 5009  cnq 9251  cltq 9257  cnp 9258

This theorem is referenced by:  prub  9393  addclprlem1  9415  mulclprlem  9418  distrlem4pr  9425  1idpr  9428  psslinpr  9430  prlem934  9432  ltaddpr  9433  ltexprlem2  9436  ltexprlem3  9437  ltexprlem6  9440  prlem936  9446  reclem2pr  9447  suplem1pr  9451

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  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-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-ltnq 9317  df-np 9380