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Theorem dedekindicclemub 13738
Description: Lemma for dedekindicc 13744. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
Hypotheses
Ref Expression
dedekindicc.a (𝜑𝐴 ∈ ℝ)
dedekindicc.b (𝜑𝐵 ∈ ℝ)
dedekindicc.lss (𝜑𝐿 ⊆ (𝐴[,]𝐵))
dedekindicc.uss (𝜑𝑈 ⊆ (𝐴[,]𝐵))
dedekindicc.lm (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)
dedekindicc.um (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)
dedekindicc.lr (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))
dedekindicc.ur (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))
dedekindicc.disj (𝜑 → (𝐿𝑈) = ∅)
dedekindicc.loc (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))
Assertion
Ref Expression
dedekindicclemub (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝐿 𝑦 < 𝑥)
Distinct variable groups:   𝐴,𝑞,𝑟,𝑦   𝑥,𝐴,𝑦   𝐵,𝑞,𝑟,𝑦   𝑥,𝐵   𝐿,𝑞,𝑦   𝑥,𝐿   𝑈,𝑞,𝑟,𝑦   𝜑,𝑞,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑟)   𝑈(𝑥)   𝐿(𝑟)

Proof of Theorem dedekindicclemub
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 dedekindicc.um . . 3 (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)
2 eleq1w 2238 . . . 4 (𝑟 = 𝑎 → (𝑟𝑈𝑎𝑈))
32cbvrexv 2704 . . 3 (∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈 ↔ ∃𝑎 ∈ (𝐴[,]𝐵)𝑎𝑈)
41, 3sylib 122 . 2 (𝜑 → ∃𝑎 ∈ (𝐴[,]𝐵)𝑎𝑈)
5 simprl 529 . . 3 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → 𝑎 ∈ (𝐴[,]𝐵))
6 dedekindicc.a . . . . 5 (𝜑𝐴 ∈ ℝ)
76adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → 𝐴 ∈ ℝ)
8 dedekindicc.b . . . . 5 (𝜑𝐵 ∈ ℝ)
98adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → 𝐵 ∈ ℝ)
10 dedekindicc.lss . . . . 5 (𝜑𝐿 ⊆ (𝐴[,]𝐵))
1110adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → 𝐿 ⊆ (𝐴[,]𝐵))
12 dedekindicc.uss . . . . 5 (𝜑𝑈 ⊆ (𝐴[,]𝐵))
1312adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → 𝑈 ⊆ (𝐴[,]𝐵))
14 dedekindicc.lm . . . . 5 (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)
1514adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)
161adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)
17 dedekindicc.lr . . . . 5 (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))
1817adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))
19 dedekindicc.ur . . . . 5 (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))
2019adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))
21 dedekindicc.disj . . . . 5 (𝜑 → (𝐿𝑈) = ∅)
2221adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → (𝐿𝑈) = ∅)
23 dedekindicc.loc . . . . 5 (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))
2423adantr 276 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))
25 simprr 531 . . . 4 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → 𝑎𝑈)
267, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25dedekindicclemuub 13737 . . 3 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → ∀𝑦𝐿 𝑦 < 𝑎)
27 brralrspcev 4058 . . 3 ((𝑎 ∈ (𝐴[,]𝐵) ∧ ∀𝑦𝐿 𝑦 < 𝑎) → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝐿 𝑦 < 𝑥)
285, 26, 27syl2anc 411 . 2 ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎𝑈)) → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝐿 𝑦 < 𝑥)
294, 28rexlimddv 2599 1 (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝐿 𝑦 < 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2148  wral 2455  wrex 2456  cin 3128  wss 3129  c0 3422   class class class wbr 4000  (class class class)co 5868  cr 7788   < clt 7969  [,]cicc 9865
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-po 4292  df-iso 4293  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-icc 9869
This theorem is referenced by: (None)
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