Theorem ndmordi 4051

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 (_ (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 3983	. . . . 5 |- (CFB) e. V
2		ndmordi.4	. . . . 5 |- R (_ (S X. S)
3	1, 2	brel 3223	. . . 4 |- ((CFA)R(CFB) -> ((CFA) e. S /\ (CFB) e. S))
4	3	pm3.26d 321	. . 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 4046	. . . 4 |- ((CFA) e. S -> (C e. S /\ A e. S))
9	8	pm3.26d 321	. . 3 |- ((CFA) e. S -> C e. S)
10	4, 9	syl 10	. 2 |- ((CFA)R(CFB) -> C e. S)
11		ndmordi.6	. . 3 |- (C e. S -> (ARB <-> (CFA)R(CFB)))
12	11	biimprd 154	. 2 |- (C e. S -> ((CFA)R(CFB) -> ARB))
13	10, 12	mpcom 49	1 |- ((CFA)R(CFB) -> ARB)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047  (/)c0 2280   class class class wbr 2619   X. cxp 3168  dom cdm 3170  (class class class)co 3963

This theorem is referenced by:  ltsopq 5075  ltexprlem4 5145  ltsosr 5203

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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