supxrbnd - Metamath Proof Explorer

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Theorem supxrbnd 12724

Description: The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)

**Assertion**
Ref	Expression
supxrbnd	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)

Proof of Theorem supxrbnd Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ressxr 10687	. . . . 5 ⊢ ℝ ⊆ ℝ^*
2		sstr 3977	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
3	1, 2	mpan2 689	. . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
4		supxrcl 12711	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
5		pnfxr 10697	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
6		xrltne 12559	. . . . . . . . . 10 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ +∞ ∈ ℝ^* ∧ sup(𝐴, ℝ^, < ) < +∞) → +∞ ≠ sup(𝐴, ℝ^, < ))
7	5, 6	mp3an2 1445	. . . . . . . . 9 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → +∞ ≠ sup(𝐴, ℝ^, < ))
8	7	necomd 3073	. . . . . . . 8 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ≠ +∞)
9	8	ex 415	. . . . . . 7 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) < +∞ → sup(𝐴, ℝ^, < ) ≠ +∞))
10	4, 9	syl 17	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^, < ) < +∞ → sup(𝐴, ℝ^, < ) ≠ +∞))
11		supxrunb2 12716	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ sup(𝐴, ℝ^*, < ) = +∞))
12		ssel2 3964	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
13	12	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
14		rexr 10689	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
15	14	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16		xrlenlt 10708	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑦))
17	16	con2bid 357	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
18	13, 15, 17	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
19	18	rexbidva 3298	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥))
20		rexnal 3240	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
21	19, 20	syl6bb 289	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
22	21	ralbidva 3198	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
23	11, 22	bitr3d 283	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) = +∞ ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
24		ralnex 3238	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
25	23, 24	syl6bb 289	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) = +∞ ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
26	25	necon2abid 3060	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) ≠ +∞))
27	10, 26	sylibrd 261	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) < +∞ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
28	27	imp 409	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
29	3, 28	sylan 582	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
30	29	3adant2 1127	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
31		supxrre 12723	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) = sup(𝐴, ℝ, < ))
32		suprcl 11603	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ)
33	31, 32	eqeltrd 2915	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
34	30, 33	syld3an3 1405	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 supcsup 8906 ℝcr 10538 +∞cpnf 10674 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875

This theorem is referenced by: supxrgtmnf 12725 ovolunlem1 24100 uniioombllem1 24184

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