Theorem suprleubex 8914

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 8040	. . . . . . . 8 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2	1	adantl 277	. . . . . . 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 7000	. . . . . 6 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
5		suprlubex.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6	4, 5	lenltd 8078	. . . . 5 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < )))
7		suprubex.ss	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
8	3, 7, 5	suprnubex 8913	. . . . 5 ⊢ (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧))
9	6, 8	bitrd 188	. . . 4 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧))
10		breq2 4009	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝐵 < 𝑤 ↔ 𝐵 < 𝑧))
11	10	notbid 667	. . . . 5 ⊢ (𝑤 = 𝑧 → (¬ 𝐵 < 𝑤 ↔ ¬ 𝐵 < 𝑧))
12	11	cbvralv 2705	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)
13	9, 12	bitr4di 198	. . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤))
14	7	sselda 3157	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
15	5	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝐵 ∈ ℝ)
16	14, 15	lenltd 8078	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤))
17	16	ralbidva 2473	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤))
18	13, 17	bitr4d 191	. 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵))
19		breq1 4008	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝐵 ↔ 𝑧 ≤ 𝐵))
20	19	cbvralv 2705	. 2 ⊢ (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)
21	18, 20	bitrdi 196	1 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456   ⊆ wss 3131   class class class wbr 4005  supcsup 6984  ℝcr 7813   < clt 7995   ≤ cle 7996

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929

This theorem depends on definitions:  df-bi 117  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-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-po 4298  df-iso 4299  df-xp 4634  df-cnv 4636  df-iota 5180  df-riota 5834  df-sup 6986  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001

This theorem is referenced by:  suprzclex  9354  suplociccex  14264