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Theorem ltordlem 11169
 Description: Lemma for ltord1 11170. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ltord.1 (𝑥 = 𝑦𝐴 = 𝐵)
ltord.2 (𝑥 = 𝐶𝐴 = 𝑀)
ltord.3 (𝑥 = 𝐷𝐴 = 𝑁)
ltord.4 𝑆 ⊆ ℝ
ltord.5 ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)
ltord.6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))
Assertion
Ref Expression
ltordlem ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem ltordlem
StepHypRef Expression
1 ltord.6 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))
21ralrimivva 3156 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵))
3 breq1 5036 . . . 4 (𝑥 = 𝐶 → (𝑥 < 𝑦𝐶 < 𝑦))
4 ltord.2 . . . . 5 (𝑥 = 𝐶𝐴 = 𝑀)
54breq1d 5043 . . . 4 (𝑥 = 𝐶 → (𝐴 < 𝐵𝑀 < 𝐵))
63, 5imbi12d 348 . . 3 (𝑥 = 𝐶 → ((𝑥 < 𝑦𝐴 < 𝐵) ↔ (𝐶 < 𝑦𝑀 < 𝐵)))
7 breq2 5037 . . . 4 (𝑦 = 𝐷 → (𝐶 < 𝑦𝐶 < 𝐷))
8 eqeq1 2802 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐷𝑦 = 𝐷))
9 ltord.1 . . . . . . . 8 (𝑥 = 𝑦𝐴 = 𝐵)
109eqeq1d 2800 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 = 𝑁𝐵 = 𝑁))
118, 10imbi12d 348 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 = 𝐷𝐴 = 𝑁) ↔ (𝑦 = 𝐷𝐵 = 𝑁)))
12 ltord.3 . . . . . 6 (𝑥 = 𝐷𝐴 = 𝑁)
1311, 12chvarvv 2005 . . . . 5 (𝑦 = 𝐷𝐵 = 𝑁)
1413breq2d 5045 . . . 4 (𝑦 = 𝐷 → (𝑀 < 𝐵𝑀 < 𝑁))
157, 14imbi12d 348 . . 3 (𝑦 = 𝐷 → ((𝐶 < 𝑦𝑀 < 𝐵) ↔ (𝐶 < 𝐷𝑀 < 𝑁)))
166, 15rspc2v 3581 . 2 ((𝐶𝑆𝐷𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵) → (𝐶 < 𝐷𝑀 < 𝑁)))
172, 16mpan9 510 1 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3882   class class class wbr 5033  ℝcr 10540   < clt 10679 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-un 3887  df-sn 4528  df-pr 4530  df-op 4534  df-br 5034 This theorem is referenced by:  ltord1  11170
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