supxrre2 - Metamath Proof Explorer

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Theorem supxrre2 13234

Description: The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)

**Assertion**
Ref	Expression
supxrre2	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))

**Proof of Theorem supxrre2**
Step	Hyp	Ref	Expression
1		supxrre1 13233	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ sup(𝐴, ℝ^, < ) < +∞))
2		ressxr 11165	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
3		sstr 3939	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
4	2, 3	mpan2 691	. . . . 5 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
5		supxrcl 13218	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
6		nltpnft 13067	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
7	4, 5, 6	3syl 18	. . . 4 ⊢ (𝐴 ⊆ ℝ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
8	7	necon2abid 2971	. . 3 ⊢ (𝐴 ⊆ ℝ → (sup(𝐴, ℝ^, < ) < +∞ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))
9	8	adantr 480	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) < +∞ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))
10	1, 9	bitrd 279	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ⊆ wss 3898 ∅c0 4282 class class class wbr 5095 supcsup 9333 ℝcr 11014 +∞cpnf 11152 ℝ^*cxr 11154 < clt 11155

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 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-pre-sup 11093

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 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-er 8630 df-en 8878 df-dom 8879 df-sdom 8880 df-sup 9335 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356

This theorem is referenced by: ovollb2 25420 nmorepnf 30752 nmoprepnf 31851 nmfnrepnf 31864

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