Theorem ndmordi 4984

Description: Elimination of redundant antecedent in an ordering law.

Hypotheses

Ref	Expression
ndmordi.3	|- A e. _V
ndmordi.2	|- dom F = (S X. S)
ndmordi.4	|- R C_ (S X. S)
ndmordi.5	|- -. (/) e. S
ndmordi.6	|- (C e. S -> (ARB <-> (CFA)R(CFB)))

Assertion

Ref	Expression
ndmordi	|- ((CFA)R(CFB) -> ARB)

Proof of Theorem ndmordi

Step	Hyp	Ref	Expression
1		oprex 4907	. . . . 5 |- (CFB) e. _V
2		ndmordi.4	. . . . 5 |- R C_ (S X. S)
3	1, 2	brel 4048	. . . 4 |- ((CFA)R(CFB) -> ((CFA) e. S /\ (CFB) e. S))
4	3	simplld 348	. . 3 |- ((CFA)R(CFB) -> (CFA) e. S)
5		ndmordi.3	. . . . 5 |- A e. _V
6		ndmordi.2	. . . . 5 |- dom F = (S X. S)
7		ndmordi.5	. . . . 5 |- -. (/) e. S
8	5, 6, 7	ndmoprrcl 4979	. . . 4 |- ((CFA) e. S -> (C e. S /\ A e. S))
9	8	simplld 348	. . 3 |- ((CFA) e. S -> C e. S)
10	4, 9	syl 12	. 2 |- ((CFA)R(CFB) -> C e. S)
11		ndmordi.6	. . 3 |- (C e. S -> (ARB <-> (CFA)R(CFB)))
12	11	biimprd 171	. 2 |- (C e. S -> ((CFA)R(CFB) -> ARB))
13	10, 12	mpcom 60	1 |- ((CFA)R(CFB) -> ARB)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  (/)c0 2875   class class class wbr 3338   X. cxp 3984  dom cdm 3986  (class class class)co 4884

This theorem is referenced by:  ltsopq 6227  ltexprlem4 6297  ltsosr 6355

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886

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