supxrbnd - Metamath Proof Explorer

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Theorem supxrbnd 13372

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 11271	. . . . 5 ⊢ ℝ ⊆ ℝ^*
2		sstr 3948	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
3	1, 2	mpan2 704	. . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
4		supxrcl 13359	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
5		pnfxr 11281	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
6		xrltne 13206	. . . . . . . . . 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 3016	. . . . . . . 8 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ≠ +∞)
9	8	ex 418	. . . . . . 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 13364	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ sup(𝐴, ℝ^*, < ) = +∞))
12		ssel2 3935	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
13	12	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
14		rexr 11273	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
15	14	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
16		xrlenlt 11292	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
19	18	rexbidva 3190	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥))
20		rexnal 3120	. . . . . . . . . . 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 3189	. . . . . . . . 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 3094	. . . . . . . 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 3003	. . . . . 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 412	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ sup(𝐴, ℝ^*, < ) < +∞) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
29	3, 28	sylan 592	. . 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 13371	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ^*, < ) = sup(𝐴, ℝ, < ))
32		suprcl 12193	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ)
33	31, 32	eqeltrd 2866	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 ∃wrex 3092 ⊆ wss 3908 ∅c0 4289 class class class wbr 5114 supcsup 9410 ℝcr 11117 +∞cpnf 11258 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-po 5574 df-so 5575 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462

This theorem is used by: supxrgtmnf 13373 ovolunlem1 25693 uniioombllem1 25777

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