Theorem ndmord 3064

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	. . . 4 |- ((A e. S /\ B e. S /\ C e. S) -> (ARB <-> (CFA)R(CFB)))
2	1	3exp 611	. . 3 |- (A e. S -> (B e. S -> (C e. S -> (ARB <-> (CFA)R(CFB)))))
3	2	imp 277	. 2 |- ((A e. S /\ B e. S) -> (C e. S -> (ARB <-> (CFA)R(CFB))))
4		ndmopr.1	. . . . 5 |- B e. V
5		ndmord.4	. . . . 5 |- R (_ (S X. S)
6	4, 5	brel 2459	. . . 4 |- (ARB -> (A e. S /\ B e. S))
7		oprex 3018	. . . . . 6 |- (CFB) e. V
8	7, 5	brel 2459	. . . . 5 |- ((CFA)R(CFB) -> ((CFA) e. S /\ (CFB) e. S))
9		ndmord.3	. . . . . . . 8 |- A e. V
10		ndmopr.2	. . . . . . . 8 |- dom F = (S X. S)
11		ndmord.5	. . . . . . . 8 |- -. (/) e. S
12	9, 10, 11	ndmoprrcl 3060	. . . . . . 7 |- ((CFA) e. S -> (C e. S /\ A e. S))
13	12	pm3.27d 262	. . . . . 6 |- ((CFA) e. S -> A e. S)
14	4, 10, 11	ndmoprrcl 3060	. . . . . . 7 |- ((CFB) e. S -> (C e. S /\ B e. S))
15	14	pm3.27d 262	. . . . . 6 |- ((CFB) e. S -> B e. S)
16	13, 15	anim12i 268	. . . . 5 |- (((CFA) e. S /\ (CFB) e. S) -> (A e. S /\ B e. S))
17	8, 16	syl 12	. . . 4 |- ((CFA)R(CFB) -> (A e. S /\ B e. S))
18	6, 17	pm5.21ni 503	. . 3 |- (-. (A e. S /\ B e. S) -> (ARB <-> (CFA)R(CFB)))
19	18	a1d 14	. 2 |- (-. (A e. S /\ B e. S) -> (C e. S -> (ARB <-> (CFA)R(CFB))))
20	3, 19	pm2.61i 110	1 |- (C e. S -> (ARB <-> (CFA)R(CFB)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  (/)c0 1707   class class class wbr 2054   X. cxp 2408  dom cdm 2410  (class class class)co 3001

This theorem is referenced by:  ltapi 3824  ltmpi 3825  ltapq 3870  ltmpq 3871  ltapr 3945  ltasr 4003

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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