Theorem ndmovordi 7618

Description: Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)

Hypotheses

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

Assertion

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

Proof of Theorem ndmovordi

Step	Hyp	Ref	Expression
1		ndmovordi.4	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
2	1	brel 5747	. . . 4 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → ((𝐶𝐹𝐴) ∈ 𝑆 ∧ (𝐶𝐹𝐵) ∈ 𝑆))
3	2	simpld 493	. . 3 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → (𝐶𝐹𝐴) ∈ 𝑆)
4		ndmovordi.2	. . . . 5 ⊢ dom 𝐹 = (𝑆 × 𝑆)
5		ndmovordi.5	. . . . 5 ⊢ ¬ ∅ ∈ 𝑆
6	4, 5	ndmovrcl 7613	. . . 4 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → (𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
7	6	simpld 493	. . 3 ⊢ ((𝐶𝐹𝐴) ∈ 𝑆 → 𝐶 ∈ 𝑆)
8	3, 7	syl 17	. 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐶 ∈ 𝑆)
9		ndmovordi.6	. . 3 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
10	9	biimprd 247	. 2 ⊢ (𝐶 ∈ 𝑆 → ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵))
11	8, 10	mpcom 38	1 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098   ⊆ wss 3949  ∅c0 4326   class class class wbr 5152   × cxp 5680  dom cdm 5682  (class class class)co 7426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-dm 5692  df-iota 6505  df-fv 6561  df-ov 7429

This theorem is referenced by:  ltexprlem4  11070  ltsosr  11125