supxrbnd - Metamath Proof Explorer

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Theorem supxrbnd 12030

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 9962	. . . . 5 ⊢ ℝ ⊆ ℝ^*
2		sstr 3576	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
3	1, 2	mpan2 703	. . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
4		supxrcl 12017	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
5		pnfxr 9971	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
6		xrltne 11870	. . . . . . . . . 10 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ +∞ ∈ ℝ^* ∧ sup(𝐴, ℝ^, < ) < +∞) → +∞ ≠ sup(𝐴, ℝ^, < ))
7	5, 6	mp3an2 1404	. . . . . . . . 9 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → +∞ ≠ sup(𝐴, ℝ^, < ))
8	7	necomd 2837	. . . . . . . 8 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ≠ +∞)
9	8	ex 449	. . . . . . 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 12022	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ sup(𝐴, ℝ^*, < ) = +∞))
12		ssel2 3563	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
13	12	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
14		rexr 9964	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
15	14	ad2antlr 759	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16		xrlenlt 9982	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦 ≤ 𝑥 ↔ ¬ 𝑥 < 𝑦))
17	16	con2bid 343	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
18	13, 15, 17	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
19	18	rexbidva 3031	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥))
20		rexnal 2978	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
21	19, 20	syl6bb 275	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
22	21	ralbidva 2968	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
23	11, 22	bitr3d 269	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) = +∞ ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
24		ralnex 2975	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
25	23, 24	syl6bb 275	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) = +∞ ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
26	25	necon2abid 2824	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) ≠ +∞))
27	10, 26	sylibrd 248	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) < +∞ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
28	27	imp 444	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
29	3, 28	sylan 487	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
30	29	3adant2 1073	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
31		supxrre 12029	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) = sup(𝐴, ℝ, < ))
32		suprcl 10862	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ)
33	31, 32	eqeltrd 2688	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
34	30, 33	syld3an3 1363	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 supcsup 8229 ℝcr 9814 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148

This theorem is referenced by: supxrgtmnf 12031 ovolunlem1 23072 uniioombllem1 23155

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