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Theorem ltordlem 11357
Description: Lemma for ltord1 11358. (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 3112 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵))
3 breq1 5056 . . . 4 (𝑥 = 𝐶 → (𝑥 < 𝑦𝐶 < 𝑦))
4 ltord.2 . . . . 5 (𝑥 = 𝐶𝐴 = 𝑀)
54breq1d 5063 . . . 4 (𝑥 = 𝐶 → (𝐴 < 𝐵𝑀 < 𝐵))
63, 5imbi12d 348 . . 3 (𝑥 = 𝐶 → ((𝑥 < 𝑦𝐴 < 𝐵) ↔ (𝐶 < 𝑦𝑀 < 𝐵)))
7 breq2 5057 . . . 4 (𝑦 = 𝐷 → (𝐶 < 𝑦𝐶 < 𝐷))
8 eqeq1 2741 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐷𝑦 = 𝐷))
9 ltord.1 . . . . . . . 8 (𝑥 = 𝑦𝐴 = 𝐵)
109eqeq1d 2739 . . . . . . 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 2007 . . . . 5 (𝑦 = 𝐷𝐵 = 𝑁)
1413breq2d 5065 . . . 4 (𝑦 = 𝐷 → (𝑀 < 𝐵𝑀 < 𝑁))
157, 14imbi12d 348 . . 3 (𝑦 = 𝐷 → ((𝐶 < 𝑦𝑀 < 𝐵) ↔ (𝐶 < 𝐷𝑀 < 𝑁)))
166, 15rspc2v 3547 . 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 1543  wcel 2110  wral 3061  wss 3866   class class class wbr 5053  cr 10728   < clt 10867
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-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-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054
This theorem is referenced by:  ltord1  11358
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