Theorem swoord1 8750

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 23	. . . 4 ⊢ (𝜑 → 𝜑)
2		swoord.6	. . . . 5 ⊢ (𝜑 → 𝐴𝑅𝐵)
3		swoer.1	. . . . . . 7 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
4		difss 4072	. . . . . . 7 ⊢ ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) ⊆ (𝑋 × 𝑋)
5	3, 4	eqsstri 3954	. . . . . 6 ⊢ 𝑅 ⊆ (𝑋 × 𝑋)
6	5	ssbri 5149	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐴(𝑋 × 𝑋)𝐵)
7		df-br 5103	. . . . . 6 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
8		opelxp1 5692	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴 ∈ 𝑋)
9	7, 8	sylbi 220	. . . . 5 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → 𝐴 ∈ 𝑋)
10	2, 6, 9	3syl 19	. . . 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 5568	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 < 𝐶 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
15	1, 10, 11, 12, 14	syl13anc 1399	. . 3 ⊢ (𝜑 → (𝐴 < 𝐶 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
16	3	brdifun 8748	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
17	10, 12, 16	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
18	2, 17	mpbid 235	. . . . 5 ⊢ (𝜑 → ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
19		orc 881	. . . . 5 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
20	18, 19	nsyl 141	. . . 4 ⊢ (𝜑 → ¬ 𝐴 < 𝐵)
21		biorf 950	. . . 4 ⊢ (¬ 𝐴 < 𝐵 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
22	20, 21	syl 18	. . 3 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐶)))
23	15, 22	sylibrd 262	. 2 ⊢ (𝜑 → (𝐴 < 𝐶 → 𝐵 < 𝐶))
24	13	swopolem 5568	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
25	1, 12, 11, 10, 24	syl13anc 1399	. . 3 ⊢ (𝜑 → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
26		olc 882	. . . . 5 ⊢ (𝐵 < 𝐴 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
27	18, 26	nsyl 141	. . . 4 ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
28		biorf 950	. . . 4 ⊢ (¬ 𝐵 < 𝐴 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
29	27, 28	syl 18	. . 3 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
30	25, 29	sylibrd 262	. 2 ⊢ (𝜑 → (𝐵 < 𝐶 → 𝐴 < 𝐶))
31	23, 30	impbid 215	1 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ∖ cdif 4045   ∪ cun 4149  ⟨cop 4589   class class class wbr 5102   × cxp 5648  ^◡ccnv 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658

This theorem is used by: (None)