Theorem suprubex 9066

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 3205	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4		lttri3 8194	. . . 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 7133	. 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
8	5, 6	supubti 7134	. . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵))
9	2, 8	mpd 13	. 2 ⊢ (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵)
10	3, 7, 9	nltled 8235	1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489   ⊆ wss 3177   class class class wbr 4062  supcsup 7117  ℝcr 7966   < clt 8149   ≤ cle 8150

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-pre-ltirr 8079  ax-pre-apti 8082

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-xp 4702  df-cnv 4704  df-iota 5254  df-riota 5927  df-sup 7119  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155

This theorem is referenced by:  suprzclex  9513