Theorem axcontlem3 27057

Description: Lemma for axcont 27067. Given the separation assumption, 𝐵 is a subset of 𝐷. (Contributed by Scott Fenton, 18-Jun-2013.)

Hypothesis

Ref	Expression
axcontlem3.1	⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}

Assertion

Ref	Expression
axcontlem3	⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) → 𝐵 ⊆ 𝐷)

Distinct variable groups:   𝐴,𝑝,𝑥   𝐵,𝑝,𝑥,𝑦   𝑁,𝑝,𝑥,𝑦   𝑈,𝑝,𝑥,𝑦   𝑍,𝑝,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐷(𝑥,𝑦,𝑝)

Proof of Theorem axcontlem3

Step	Hyp	Ref	Expression
1		simplr2 1218	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) → 𝐵 ⊆ (𝔼‘𝑁))
2		simpr2 1197	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) → 𝑈 ∈ 𝐴)
3	2	anim1i 618	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) ∧ 𝑝 ∈ 𝐵) → (𝑈 ∈ 𝐴 ∧ 𝑝 ∈ 𝐵))
4		simplr3 1219	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)
5	4	adantr 484	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) ∧ 𝑝 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)
6		breq1 5056	. . . . . 6 ⊢ (𝑥 = 𝑈 → (𝑥 Btwn ⟨𝑍, 𝑦⟩ ↔ 𝑈 Btwn ⟨𝑍, 𝑦⟩))
7		opeq2 4785	. . . . . . 7 ⊢ (𝑦 = 𝑝 → ⟨𝑍, 𝑦⟩ = ⟨𝑍, 𝑝⟩)
8	7	breq2d 5065	. . . . . 6 ⊢ (𝑦 = 𝑝 → (𝑈 Btwn ⟨𝑍, 𝑦⟩ ↔ 𝑈 Btwn ⟨𝑍, 𝑝⟩))
9	6, 8	rspc2v 3547	. . . . 5 ⊢ ((𝑈 ∈ 𝐴 ∧ 𝑝 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩ → 𝑈 Btwn ⟨𝑍, 𝑝⟩))
10	3, 5, 9	sylc 65	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) ∧ 𝑝 ∈ 𝐵) → 𝑈 Btwn ⟨𝑍, 𝑝⟩)
11	10	orcd 873	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) ∧ 𝑝 ∈ 𝐵) → (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩))
12	11	ralrimiva 3105	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) → ∀𝑝 ∈ 𝐵 (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩))
13		axcontlem3.1	. . . 4 ⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}
14	13	sseq2i 3930	. . 3 ⊢ (𝐵 ⊆ 𝐷 ↔ 𝐵 ⊆ {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)})
15		ssrab 3986	. . 3 ⊢ (𝐵 ⊆ {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)} ↔ (𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑝 ∈ 𝐵 (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)))
16	14, 15	bitri 278	. 2 ⊢ (𝐵 ⊆ 𝐷 ↔ (𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑝 ∈ 𝐵 (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)))
17	1, 12, 16	sylanbrc 586	1 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) → 𝐵 ⊆ 𝐷)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  {crab 3065   ⊆ wss 3866  ⟨cop 4547   class class class wbr 5053  ‘cfv 6380  ℕcn 11830  𝔼cee 26979   Btwn cbtwn 26980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054

This theorem is referenced by:  axcontlem9  27063  axcontlem10  27064