Theorem ndmord 4064

Description: Elimination of redundant antecedents in an ordering law.

Hypotheses

Ref	Expression
ndmopr.1	|- B e. V
ndmopr.2	|- dom F = (S X. S)
ndmord.3	|- A e. V
ndmord.4	|- R (_ (S X. S)
ndmord.5	|- -. (/) e. S
ndmord.6	|- ((A e. S /\ B e. S /\ C e. S) -> (ARB <-> (CFA)R(CFB)))

Assertion

Ref	Expression
ndmord	|- (C e. S -> (ARB <-> (CFA)R(CFB)))

Proof of Theorem ndmord

Step	Hyp	Ref	Expression
1		ndmord.6	. . 3 |- ((A e. S /\ B e. S /\ C e. S) -> (ARB <-> (CFA)R(CFB)))
2	1	3expia 839	. 2 |- ((A e. S /\ B e. S) -> (C e. S -> (ARB <-> (CFA)R(CFB))))
3		ndmopr.1	. . . . 5 |- B e. V
4		ndmord.4	. . . . 5 |- R (_ (S X. S)
5	3, 4	brel 3237	. . . 4 |- (ARB -> (A e. S /\ B e. S))
6		oprex 3997	. . . . . 6 |- (CFB) e. V
7	6, 4	brel 3237	. . . . 5 |- ((CFA)R(CFB) -> ((CFA) e. S /\ (CFB) e. S))
8		ndmord.3	. . . . . . . 8 |- A e. V
9		ndmopr.2	. . . . . . . 8 |- dom F = (S X. S)
10		ndmord.5	. . . . . . . 8 |- -. (/) e. S
11	8, 9, 10	ndmoprrcl 4060	. . . . . . 7 |- ((CFA) e. S -> (C e. S /\ A e. S))
12	11	pm3.27d 325	. . . . . 6 |- ((CFA) e. S -> A e. S)
13	3, 9, 10	ndmoprrcl 4060	. . . . . . 7 |- ((CFB) e. S -> (C e. S /\ B e. S))
14	13	pm3.27d 325	. . . . . 6 |- ((CFB) e. S -> B e. S)
15	12, 14	anim12i 333	. . . . 5 |- (((CFA) e. S /\ (CFB) e. S) -> (A e. S /\ B e. S))
16	7, 15	syl 10	. . . 4 |- ((CFA)R(CFB) -> (A e. S /\ B e. S))
17	5, 16	pm5.21ni 682	. . 3 |- (-. (A e. S /\ B e. S) -> (ARB <-> (CFA)R(CFB)))
18	17	a1d 12	. 2 |- (-. (A e. S /\ B e. S) -> (C e. S -> (ARB <-> (CFA)R(CFB))))
19	2, 18	pm2.61i 126	1 |- (C e. S -> (ARB <-> (CFA)R(CFB)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 779   = wceq 960   e. wcel 962  Vcvv 1818   (_ wss 2056  (/)c0 2289   class class class wbr 2632   X. cxp 3182  dom cdm 3184  (class class class)co 3977

This theorem is referenced by:  ltapi 5043  ltmpi 5044  ltapq 5089  ltmpq 5090  ltapr 5164  ltasr 5222

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-xp 3198  df-cnv 3200  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fv 3212  df-opr 3979

Copyright terms: Public domain