Theorem iccsupr 10201

Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.)

Assertion

Ref	Expression
iccsupr	⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem iccsupr

Step	Hyp	Ref	Expression
1		iccssre 10190	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
2		sstr 3235	. . . . 5 ⊢ ((𝑆 ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ⊆ ℝ) → 𝑆 ⊆ ℝ)
3	2	ancoms 268	. . . 4 ⊢ (((𝐴[,]𝐵) ⊆ ℝ ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ)
4	1, 3	sylan 283	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝑆 ⊆ ℝ)
5	4	3adant3 1043	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → 𝑆 ⊆ ℝ)
6		ne0i 3501	. . 3 ⊢ (𝐶 ∈ 𝑆 → 𝑆 ≠ ∅)
7	6	3ad2ant3 1046	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → 𝑆 ≠ ∅)
8		simplr 529	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
9		ssel 3221	. . . . . . . 8 ⊢ (𝑆 ⊆ (𝐴[,]𝐵) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (𝐴[,]𝐵)))
10		elicc2 10173	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
11	10	biimpd 144	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
12	9, 11	sylan9r 410	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → (𝑦 ∈ 𝑆 → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
13	12	imp 124	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦 ∈ 𝑆) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))
14	13	simp3d 1037	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ≤ 𝐵)
15	14	ralrimiva 2605	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵)
16		breq2 4092	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝐵))
17	16	ralbidv 2532	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵))
18	17	rspcev 2910	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)
19	8, 15, 18	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)
20	19	3adant3 1043	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)
21	5, 7, 20	3jca 1203	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ∀wral 2510  ∃wrex 2511   ⊆ wss 3200  ∅c0 3494   class class class wbr 4088  (class class class)co 6018  ℝcr 8031   ≤ cle 8215  [,]cicc 10126

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-icc 10130

This theorem is referenced by: (None)