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Theorem ltordlem 8256
Description: Lemma for eqord1 8257. (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 2514 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵))
3 breq1 3932 . . . 4 (𝑥 = 𝐶 → (𝑥 < 𝑦𝐶 < 𝑦))
4 ltord.2 . . . . 5 (𝑥 = 𝐶𝐴 = 𝑀)
54breq1d 3939 . . . 4 (𝑥 = 𝐶 → (𝐴 < 𝐵𝑀 < 𝐵))
63, 5imbi12d 233 . . 3 (𝑥 = 𝐶 → ((𝑥 < 𝑦𝐴 < 𝐵) ↔ (𝐶 < 𝑦𝑀 < 𝐵)))
7 breq2 3933 . . . 4 (𝑦 = 𝐷 → (𝐶 < 𝑦𝐶 < 𝐷))
8 eqeq1 2146 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐷𝑦 = 𝐷))
9 ltord.1 . . . . . . . 8 (𝑥 = 𝑦𝐴 = 𝐵)
109eqeq1d 2148 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 = 𝑁𝐵 = 𝑁))
118, 10imbi12d 233 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 = 𝐷𝐴 = 𝑁) ↔ (𝑦 = 𝐷𝐵 = 𝑁)))
12 ltord.3 . . . . . 6 (𝑥 = 𝐷𝐴 = 𝑁)
1311, 12chvarv 1909 . . . . 5 (𝑦 = 𝐷𝐵 = 𝑁)
1413breq2d 3941 . . . 4 (𝑦 = 𝐷 → (𝑀 < 𝐵𝑀 < 𝑁))
157, 14imbi12d 233 . . 3 (𝑦 = 𝐷 → ((𝐶 < 𝑦𝑀 < 𝐵) ↔ (𝐶 < 𝐷𝑀 < 𝑁)))
166, 15rspc2v 2802 . 2 ((𝐶𝑆𝐷𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵) → (𝐶 < 𝐷𝑀 < 𝑁)))
172, 16mpan9 279 1 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  wral 2416  wss 3071   class class class wbr 3929  cr 7631   < clt 7812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930
This theorem is referenced by:  eqord1  8257
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