Theorem ndmordi 3065

Description: Elimination of redundant antecedent in an ordering law.

Hypotheses

Ref	Expression
ndmordi.3	⊢ A ∈ V
ndmordi.2	⊢ dom F = (S × S)
ndmordi.4	⊢ R ⊆ (S × S)
ndmordi.5	⊢ ¬ ∅ ∈ S
ndmordi.6	⊢ (C ∈ S → (ARB ↔ (CFA)R(CFB)))

Assertion

Ref	Expression
ndmordi	⊢ ((CFA)R(CFB) → ARB)

Proof of Theorem ndmordi

Step	Hyp	Ref	Expression
1		oprex 3018	. . . . 5 ⊢ (CFB) ∈ V
2		ndmordi.4	. . . . 5 ⊢ R ⊆ (S × S)
3	1, 2	brel 2459	. . . 4 ⊢ ((CFA)R(CFB) → ((CFA) ∈ S ∧ (CFB) ∈ S))
4	3	pm3.26d 258	. . 3 ⊢ ((CFA)R(CFB) → (CFA) ∈ S)
5		ndmordi.3	. . . . 5 ⊢ A ∈ V
6		ndmordi.2	. . . . 5 ⊢ dom F = (S × S)
7		ndmordi.5	. . . . 5 ⊢ ¬ ∅ ∈ S
8	5, 6, 7	ndmoprrcl 3060	. . . 4 ⊢ ((CFA) ∈ S → (C ∈ S ∧ A ∈ S))
9	8	pm3.26d 258	. . 3 ⊢ ((CFA) ∈ S → C ∈ S)
10	4, 9	syl 12	. 2 ⊢ ((CFA)R(CFB) → C ∈ S)
11		ndmordi.6	. . 3 ⊢ (C ∈ S → (ARB ↔ (CFA)R(CFB)))
12	11	biimprd 136	. 2 ⊢ (C ∈ S → ((CFA)R(CFB) → ARB))
13	10, 12	mpcom 49	1 ⊢ ((CFA)R(CFB) → ARB)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   × cxp 2408  dom cdm 2410  (class class class)co 3001

This theorem is referenced by:  ltsopq 3869  ltexprlem4 3939  ltsosr 3997

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-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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