Theorem suprubex 9275

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 3249	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4		lttri3 8399	. . . 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 7332	. 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
8	5, 6	supubti 7333	. . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵))
9	2, 8	mpd 13	. 2 ⊢ (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵)
10	3, 7, 9	nltled 8441	1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   ⊆ wss 3220   class class class wbr 4128  supcsup 7316  ℝcr 8172   < clt 8354   ≤ cle 8355

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-pre-ltirr 8285  ax-pre-apti 8288

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-cnv 4780  df-iota 5335  df-riota 6032  df-sup 7318  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360

This theorem is referenced by:  suprzclex  9727