Theorem swoord1 8777

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 4136	. . . . . . 7 ⊢ ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) ⊆ (𝑋 × 𝑋)
5	3, 4	eqsstri 4030	. . . . . 6 ⊢ 𝑅 ⊆ (𝑋 × 𝑋)
6	5	ssbri 5188	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐴(𝑋 × 𝑋)𝐵)
7		df-br 5144	. . . . . 6 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
8		opelxp1 5727	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴 ∈ 𝑋)
9	7, 8	sylbi 217	. . . . 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 5602	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 < 𝐶 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
15	1, 10, 11, 12, 14	syl13anc 1374	. . 3 ⊢ (𝜑 → (𝐴 < 𝐶 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
16	3	brdifun 8775	. . . . . . 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 232	. . . . 5 ⊢ (𝜑 → ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
19		orc 868	. . . . 5 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
20	18, 19	nsyl 140	. . . 4 ⊢ (𝜑 → ¬ 𝐴 < 𝐵)
21		biorf 937	. . . 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 259	. 2 ⊢ (𝜑 → (𝐴 < 𝐶 → 𝐵 < 𝐶))
24	13	swopolem 5602	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
25	1, 12, 11, 10, 24	syl13anc 1374	. . 3 ⊢ (𝜑 → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
26		olc 869	. . . . 5 ⊢ (𝐵 < 𝐴 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
27	18, 26	nsyl 140	. . . 4 ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
28		biorf 937	. . . 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 259	. 2 ⊢ (𝜑 → (𝐵 < 𝐶 → 𝐴 < 𝐶))
31	23, 30	impbid 212	1 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ∖ cdif 3948   ∪ cun 3949  ⟨cop 4632   class class class wbr 5143   × cxp 5683  ^◡ccnv 5684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693

This theorem is referenced by: (None)