Theorem dedekindicclemub 15509

Description: Lemma for dedekindicc 15515. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)

Hypotheses

Ref	Expression
dedekindicc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dedekindicc.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dedekindicc.lss	⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))
dedekindicc.uss	⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))
dedekindicc.lm	⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
dedekindicc.um	⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
dedekindicc.lr	⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
dedekindicc.ur	⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
dedekindicc.disj	⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅)
dedekindicc.loc	⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))

Assertion

Ref	Expression
dedekindicclemub	⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 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.

Step	Hyp	Ref	Expression
1		dedekindicc.um	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
2		eleq1w 2295	. . . 4 ⊢ (𝑟 = 𝑎 → (𝑟 ∈ 𝑈 ↔ 𝑎 ∈ 𝑈))
3	2	cbvrexv 2781	. . 3 ⊢ (∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈 ↔ ∃𝑎 ∈ (𝐴[,]𝐵)𝑎 ∈ 𝑈)
4	1, 3	sylib 122	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝐴[,]𝐵)𝑎 ∈ 𝑈)
5		simprl 531	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝑎 ∈ (𝐴[,]𝐵))
6		dedekindicc.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝐴 ∈ ℝ)
8		dedekindicc.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝐵 ∈ ℝ)
10		dedekindicc.lss	. . . . 5 ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))
11	10	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝐿 ⊆ (𝐴[,]𝐵))
12		dedekindicc.uss	. . . . 5 ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))
13	12	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝑈 ⊆ (𝐴[,]𝐵))
14		dedekindicc.lm	. . . . 5 ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
15	14	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
16	1	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
17		dedekindicc.lr	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
18	17	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
19		dedekindicc.ur	. . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
20	19	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
21		dedekindicc.disj	. . . . 5 ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅)
22	21	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → (𝐿 ∩ 𝑈) = ∅)
23		dedekindicc.loc	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))
24	23	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))
25		simprr 533	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝑎 ∈ 𝑈)
26	7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25	dedekindicclemuub 15508	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∀𝑦 ∈ 𝐿 𝑦 < 𝑎)
27		brralrspcev 4170	. . 3 ⊢ ((𝑎 ∈ (𝐴[,]𝐵) ∧ ∀𝑦 ∈ 𝐿 𝑦 < 𝑎) → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥)
28	5, 26, 27	syl2anc 411	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥)
29	4, 28	rexlimddv 2667	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523   ∩ cin 3212   ⊆ wss 3213  ∅c0 3510   class class class wbr 4111  (class class class)co 6052  ℝcr 8128   < clt 8310  [,]cicc 10227

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-icc 10231

This theorem is referenced by: (None)