Theorem axcontlem3 29526

Description: Lemma for axcont 29536. 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 1235	. 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 1214	. . . . . 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 627	. . . . 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 1236	. . . . . 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 486	. . . . 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 5106	. . . . . 6 ⊢ (𝑥 = 𝑈 → (𝑥 Btwn ⟨𝑍, 𝑦⟩ ↔ 𝑈 Btwn ⟨𝑍, 𝑦⟩))
7		opeq2 4834	. . . . . . 7 ⊢ (𝑦 = 𝑝 → ⟨𝑍, 𝑦⟩ = ⟨𝑍, 𝑝⟩)
8	7	breq2d 5115	. . . . . 6 ⊢ (𝑦 = 𝑝 → (𝑈 Btwn ⟨𝑍, 𝑦⟩ ↔ 𝑈 Btwn ⟨𝑍, 𝑝⟩))
9	6, 8	rspc2v 3587	. . . . 5 ⊢ ((𝑈 ∈ 𝐴 ∧ 𝑝 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩ → 𝑈 Btwn ⟨𝑍, 𝑝⟩))
10	3, 5, 9	sylc 66	. . . 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 887	. . 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 3155	. 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 3960	. . 3 ⊢ (𝐵 ⊆ 𝐷 ↔ 𝐵 ⊆ {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)})
15		ssrab 4019	. . 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 595	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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103  ‘cfv 6531  ℕcn 12316  𝔼cee 29447   Btwn cbtwn 29448

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104

This theorem is used by:  axcontlem9  29532  axcontlem10  29533