supxrbnd - Metamath Proof Explorer

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Theorem supxrbnd 13355

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 11254	. . . . 5 ⊢ ℝ ⊆ ℝ^*
2		sstr 3946	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
3	1, 2	mpan2 703	. . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
4		supxrcl 13342	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
5		pnfxr 11264	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
6		xrltne 13189	. . . . . . . . . 10 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ +∞ ∈ ℝ^* ∧ sup(𝐴, ℝ^, < ) < +∞) → +∞ ≠ sup(𝐴, ℝ^, < ))
7	5, 6	mp3an2 1478	. . . . . . . . 9 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → +∞ ≠ sup(𝐴, ℝ^, < ))
8	7	necomd 3013	. . . . . . . 8 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ≠ +∞)
9	8	ex 417	. . . . . . 7 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) < +∞ → sup(𝐴, ℝ^, < ) ≠ +∞))
10	4, 9	syl 18	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^, < ) < +∞ → sup(𝐴, ℝ^, < ) ≠ +∞))
11		supxrunb2 13347	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ sup(𝐴, ℝ^*, < ) = +∞))
12		ssel2 3933	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
13	12	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
14		rexr 11256	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
15	14	ad2antlr 739	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16		xrlenlt 11275	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
19	18	rexbidva 3187	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥))
20		rexnal 3117	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
21	19, 20	bitrdi 290	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
22	21	ralbidva 3186	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
23	11, 22	bitr3d 284	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) = +∞ ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
24		ralnex 3091	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
25	23, 24	bitrdi 290	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) = +∞ ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
26	25	necon2abid 3000	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) ≠ +∞))
27	10, 26	sylibrd 262	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (sup(𝐴, ℝ^*, < ) < +∞ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
28	27	imp 411	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
29	3, 28	sylan 591	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
30	29	3adant2 1149	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
31		supxrre 13354	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) = sup(𝐴, ℝ, < ))
32		suprcl 12176	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ)
33	31, 32	eqeltrd 2863	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
34	30, 33	syld3an3 1436	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 ⊆ wss 3906 ∅c0 4287 class class class wbr 5110 supcsup 9401 ℝcr 11100 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-sup 9403 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445

This theorem is referenced by: supxrgtmnf 13356 ovolunlem1 25637 uniioombllem1 25721

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