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Theorem ltordlem 10432
Description: Lemma for ltord1 10433. (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 2954 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵))
3 breq1 4586 . . . 4 (𝑥 = 𝐶 → (𝑥 < 𝑦𝐶 < 𝑦))
4 ltord.2 . . . . 5 (𝑥 = 𝐶𝐴 = 𝑀)
54breq1d 4593 . . . 4 (𝑥 = 𝐶 → (𝐴 < 𝐵𝑀 < 𝐵))
63, 5imbi12d 333 . . 3 (𝑥 = 𝐶 → ((𝑥 < 𝑦𝐴 < 𝐵) ↔ (𝐶 < 𝑦𝑀 < 𝐵)))
7 breq2 4587 . . . 4 (𝑦 = 𝐷 → (𝐶 < 𝑦𝐶 < 𝐷))
8 eqeq1 2614 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐷𝑦 = 𝐷))
9 ltord.1 . . . . . . . 8 (𝑥 = 𝑦𝐴 = 𝐵)
109eqeq1d 2612 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 = 𝑁𝐵 = 𝑁))
118, 10imbi12d 333 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 = 𝐷𝐴 = 𝑁) ↔ (𝑦 = 𝐷𝐵 = 𝑁)))
12 ltord.3 . . . . . 6 (𝑥 = 𝐷𝐴 = 𝑁)
1311, 12chvarv 2251 . . . . 5 (𝑦 = 𝐷𝐵 = 𝑁)
1413breq2d 4595 . . . 4 (𝑦 = 𝐷 → (𝑀 < 𝐵𝑀 < 𝑁))
157, 14imbi12d 333 . . 3 (𝑦 = 𝐷 → ((𝐶 < 𝑦𝑀 < 𝐵) ↔ (𝐶 < 𝐷𝑀 < 𝑁)))
166, 15rspc2v 3293 . 2 ((𝐶𝑆𝐷𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵) → (𝐶 < 𝐷𝑀 < 𝑁)))
172, 16mpan9 485 1 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  wss 3540   class class class wbr 4583  cr 9814   < clt 9953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584
This theorem is referenced by:  ltord1  10433
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