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Theorem dedekindeulemub 14367
Description: Lemma for dedekindeu 14372. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
Hypotheses
Ref Expression
dedekindeu.lss (𝜑𝐿 ⊆ ℝ)
dedekindeu.uss (𝜑𝑈 ⊆ ℝ)
dedekindeu.lm (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)
dedekindeu.um (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)
dedekindeu.lr (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))
dedekindeu.ur (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))
dedekindeu.disj (𝜑 → (𝐿𝑈) = ∅)
dedekindeu.loc (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))
Assertion
Ref Expression
dedekindeulemub (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐿 𝑦 < 𝑥)
Distinct variable groups:   𝐿,𝑞,𝑟,𝑦   𝑥,𝐿,𝑟,𝑦   𝑈,𝑞,𝑟,𝑦   𝜑,𝑞,𝑟,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝑈(𝑥)

Proof of Theorem dedekindeulemub
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 dedekindeu.um . . 3 (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)
2 eleq1w 2248 . . . 4 (𝑟 = 𝑎 → (𝑟𝑈𝑎𝑈))
32cbvrexv 2716 . . 3 (∃𝑟 ∈ ℝ 𝑟𝑈 ↔ ∃𝑎 ∈ ℝ 𝑎𝑈)
41, 3sylib 122 . 2 (𝜑 → ∃𝑎 ∈ ℝ 𝑎𝑈)
5 simprl 529 . . 3 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → 𝑎 ∈ ℝ)
6 dedekindeu.lss . . . . 5 (𝜑𝐿 ⊆ ℝ)
76adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → 𝐿 ⊆ ℝ)
8 dedekindeu.uss . . . . 5 (𝜑𝑈 ⊆ ℝ)
98adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → 𝑈 ⊆ ℝ)
10 dedekindeu.lm . . . . 5 (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)
1110adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → ∃𝑞 ∈ ℝ 𝑞𝐿)
121adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → ∃𝑟 ∈ ℝ 𝑟𝑈)
13 dedekindeu.lr . . . . 5 (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))
1413adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))
15 dedekindeu.ur . . . . 5 (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))
1615adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))
17 dedekindeu.disj . . . . 5 (𝜑 → (𝐿𝑈) = ∅)
1817adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → (𝐿𝑈) = ∅)
19 dedekindeu.loc . . . . 5 (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))
2019adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))
21 simprr 531 . . . 4 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → 𝑎𝑈)
227, 9, 11, 12, 14, 16, 18, 20, 21dedekindeulemuub 14366 . . 3 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → ∀𝑦𝐿 𝑦 < 𝑎)
23 brralrspcev 4073 . . 3 ((𝑎 ∈ ℝ ∧ ∀𝑦𝐿 𝑦 < 𝑎) → ∃𝑥 ∈ ℝ ∀𝑦𝐿 𝑦 < 𝑥)
245, 22, 23syl2anc 411 . 2 ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎𝑈)) → ∃𝑥 ∈ ℝ ∀𝑦𝐿 𝑦 < 𝑥)
254, 24rexlimddv 2609 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐿 𝑦 < 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709   = wceq 1363  wcel 2158  wral 2465  wrex 2466  cin 3140  wss 3141  c0 3434   class class class wbr 4015  cr 7823   < clt 8005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-pre-ltwlin 7937
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011
This theorem is referenced by:  dedekindeulemlub  14369
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