supxrre2 - Metamath Proof Explorer

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Theorem supxrre2 13345

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 13344	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ sup(𝐴, ℝ^, < ) < +∞))
2		ressxr 11241	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
3		sstr 3947	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
4	2, 3	mpan2 703	. . . . 5 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
5		supxrcl 13329	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
6		nltpnft 13178	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
7	4, 5, 6	3syl 19	. . . 4 ⊢ (𝐴 ⊆ ℝ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
8	7	necon2abid 3002	. . 3 ⊢ (𝐴 ⊆ ℝ → (sup(𝐴, ℝ^, < ) < +∞ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))
9	8	adantr 485	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) < +∞ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))
10	1, 9	bitrd 282	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ sup(𝐴, ℝ^, < ) ≠ +∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ⊆ wss 3907 ∅c0 4288 class class class wbr 5104 supcsup 9388 ℝcr 11087 +∞cpnf 11228 ℝ^*cxr 11230 < clt 11231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-sup 9390 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432

This theorem is referenced by: ovollb2 25605 nmorepnf 31025 nmoprepnf 32124 nmfnrepnf 32137

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