Theorem suprubex 9136

Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)

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)
suprubex.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
suprubex	⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))

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

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

Proof of Theorem suprubex

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

Step	Hyp	Ref	Expression
1		suprubex.ss	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		suprubex.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3	1, 2	sseldd 3227	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4		lttri3 8264	. . . 4 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
5	4	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
6		suprubex.ex	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
7	5, 6	supclti 7202	. 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
8	5, 6	supubti 7203	. . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵))
9	2, 8	mpd 13	. 2 ⊢ (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵)
10	3, 7, 9	nltled 8305	1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2201  ∀wral 2509  ∃wrex 2510   ⊆ wss 3199   class class class wbr 4089  supcsup 7186  ℝcr 8036   < clt 8219   ≤ cle 8220

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-pre-ltirr 8149  ax-pre-apti 8152

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-xp 4733  df-cnv 4735  df-iota 5288  df-riota 5976  df-sup 7188  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225

This theorem is referenced by:  suprzclex  9583