Theorem suprubex 9106

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 3225	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4		lttri3 8234	. . . 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 7173	. 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
8	5, 6	supubti 7174	. . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵))
9	2, 8	mpd 13	. 2 ⊢ (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵)
10	3, 7, 9	nltled 8275	1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ⊆ wss 3197   class class class wbr 4083  supcsup 7157  ℝcr 8006   < clt 8189   ≤ cle 8190

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-pre-ltirr 8119  ax-pre-apti 8122

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-iota 5278  df-riota 5960  df-sup 7159  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195

This theorem is referenced by:  suprzclex  9553