Theorem suprleubex 8512

Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)

Hypotheses

Ref	Expression
suprubex.ex	⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
suprubex.ss	⊢ (𝜑 → 𝐴 ⊆ ℝ)
suprlubex.b	⊢ (𝜑 → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
suprleubex	⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥   𝑧,𝐵

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑥,𝑦)

Proof of Theorem suprleubex

Dummy variables 𝑓 𝑔 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lttri3 7662	. . . . . . . 8 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2	1	adantl 272	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3		suprubex.ex	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
4	2, 3	supclti 6773	. . . . . 6 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
5		suprlubex.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6	4, 5	lenltd 7698	. . . . 5 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < )))
7		suprubex.ss	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
8	3, 7, 5	suprnubex 8511	. . . . 5 ⊢ (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧))
9	6, 8	bitrd 187	. . . 4 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧))
10		breq2 3871	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝐵 < 𝑤 ↔ 𝐵 < 𝑧))
11	10	notbid 630	. . . . 5 ⊢ (𝑤 = 𝑧 → (¬ 𝐵 < 𝑤 ↔ ¬ 𝐵 < 𝑧))
12	11	cbvralv 2604	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)
13	9, 12	syl6bbr 197	. . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤))
14	7	sselda 3039	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
15	5	adantr 271	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝐵 ∈ ℝ)
16	14, 15	lenltd 7698	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤))
17	16	ralbidva 2387	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤))
18	13, 17	bitr4d 190	. 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵))
19		breq1 3870	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝐵 ↔ 𝑧 ≤ 𝐵))
20	19	cbvralv 2604	. 2 ⊢ (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)
21	18, 20	syl6bb 195	1 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371   ⊆ wss 3013   class class class wbr 3867  supcsup 6757  ℝcr 7446   < clt 7619   ≤ cle 7620

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-po 4147  df-iso 4148  df-xp 4473  df-cnv 4475  df-iota 5014  df-riota 5646  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625

This theorem is referenced by:  suprzclex  8943