supxrbnd - Metamath Proof Explorer

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Theorem supxrbnd 13229

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 11163	. . . . 5 ⊢ ℝ ⊆ ℝ^*
2		sstr 3939	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
3	1, 2	mpan2 691	. . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
4		supxrcl 13216	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
5		pnfxr 11173	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
6		xrltne 13064	. . . . . . . . . 10 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ +∞ ∈ ℝ^* ∧ sup(𝐴, ℝ^, < ) < +∞) → +∞ ≠ sup(𝐴, ℝ^, < ))
7	5, 6	mp3an2 1451	. . . . . . . . 9 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → +∞ ≠ sup(𝐴, ℝ^, < ))
8	7	necomd 2984	. . . . . . . 8 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ≠ +∞)
9	8	ex 412	. . . . . . 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 13221	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ sup(𝐴, ℝ^*, < ) = +∞))
12		ssel2 3925	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
13	12	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
14		rexr 11165	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
15	14	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16		xrlenlt 11184	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑦))
17	16	con2bid 354	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
18	13, 15, 17	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
19	18	rexbidva 3155	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥))
20		rexnal 3085	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
21	19, 20	bitrdi 287	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
22	21	ralbidva 3154	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
23	11, 22	bitr3d 281	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) = +∞ ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
24		ralnex 3059	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
25	23, 24	bitrdi 287	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) = +∞ ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
26	25	necon2abid 2971	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) ≠ +∞))
27	10, 26	sylibrd 259	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) < +∞ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
28	27	imp 406	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
29	3, 28	sylan 580	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
30	29	3adant2 1131	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
31		supxrre 13228	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) = sup(𝐴, ℝ, < ))
32		suprcl 12089	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ)
33	31, 32	eqeltrd 2833	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
34	30, 33	syld3an3 1411	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 ∃wrex 3057 ⊆ wss 3898 ∅c0 4282 class class class wbr 5093 supcsup 9331 ℝcr 11012 +∞cpnf 11150 ℝ^*cxr 11152 < clt 11153 ≤ cle 11154

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-sup 9333 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354

This theorem is referenced by: supxrgtmnf 13230 ovolunlem1 25426 uniioombllem1 25510

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