Theorem swoord1 8529

Description: The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Hypotheses

Ref	Expression
swoer.1	⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
swoer.2	⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
swoer.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
swoord.4	⊢ (𝜑 → 𝐵 ∈ 𝑋)
swoord.5	⊢ (𝜑 → 𝐶 ∈ 𝑋)
swoord.6	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
swoord1	⊢ (𝜑 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶))

Distinct variable groups:   𝑥,𝑦,𝑧, <   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)

Proof of Theorem swoord1

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
2		swoord.6	. . . . 5 ⊢ (𝜑 → 𝐴𝑅𝐵)
3		swoer.1	. . . . . . 7 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
4		difss 4066	. . . . . . 7 ⊢ ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) ⊆ (𝑋 × 𝑋)
5	3, 4	eqsstri 3955	. . . . . 6 ⊢ 𝑅 ⊆ (𝑋 × 𝑋)
6	5	ssbri 5119	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐴(𝑋 × 𝑋)𝐵)
7		df-br 5075	. . . . . 6 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
8		opelxp1 5630	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴 ∈ 𝑋)
9	7, 8	sylbi 216	. . . . 5 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → 𝐴 ∈ 𝑋)
10	2, 6, 9	3syl 18	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
11		swoord.5	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
12		swoord.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
13		swoer.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
14	13	swopolem 5513	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 < 𝐶 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
15	1, 10, 11, 12, 14	syl13anc 1371	. . 3 ⊢ (𝜑 → (𝐴 < 𝐶 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
16	3	brdifun 8527	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
17	10, 12, 16	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
18	2, 17	mpbid 231	. . . . 5 ⊢ (𝜑 → ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
19		orc 864	. . . . 5 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
20	18, 19	nsyl 140	. . . 4 ⊢ (𝜑 → ¬ 𝐴 < 𝐵)
21		biorf 934	. . . 4 ⊢ (¬ 𝐴 < 𝐵 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
22	20, 21	syl 17	. . 3 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
23	15, 22	sylibrd 258	. 2 ⊢ (𝜑 → (𝐴 < 𝐶 → 𝐵 < 𝐶))
24	13	swopolem 5513	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
25	1, 12, 11, 10, 24	syl13anc 1371	. . 3 ⊢ (𝜑 → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
26		olc 865	. . . . 5 ⊢ (𝐵 < 𝐴 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
27	18, 26	nsyl 140	. . . 4 ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
28		biorf 934	. . . 4 ⊢ (¬ 𝐵 < 𝐴 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
29	27, 28	syl 17	. . 3 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
30	25, 29	sylibrd 258	. 2 ⊢ (𝜑 → (𝐵 < 𝐶 → 𝐴 < 𝐶))
31	23, 30	impbid 211	1 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ∪ cun 3885  ⟨cop 4567   class class class wbr 5074   × cxp 5587  ^◡ccnv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597

This theorem is referenced by: (None)