supxrgtmnf - Metamath Proof Explorer

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Theorem supxrgtmnf 13311

Description: The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)

**Assertion**
Ref	Expression
supxrgtmnf	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ^*, < ))

**Proof of Theorem supxrgtmnf**
Step	Hyp	Ref	Expression
1		supxrbnd 13310	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)
2	1	3expia 1118	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) < +∞ → sup(𝐴, ℝ^, < ) ∈ ℝ))
3	2	con3d 152	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (¬ sup(𝐴, ℝ^, < ) ∈ ℝ → ¬ sup(𝐴, ℝ^, < ) < +∞))
4		ressxr 11259	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
5		sstr 3985	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
6	4, 5	mpan2 688	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
7		supxrcl 13297	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → sup(𝐴, ℝ^, < ) ∈ ℝ^)
9	8	adantr 480	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
10		nltpnft 13146	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
11	9, 10	syl 17	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
12	3, 11	sylibrd 259	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (¬ sup(𝐴, ℝ^, < ) ∈ ℝ → sup(𝐴, ℝ^, < ) = +∞))
13	12	orrd 860	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ∨ sup(𝐴, ℝ^, < ) = +∞))
14		mnfltxr 13110	. 2 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∨ sup(𝐴, ℝ^, < ) = +∞) → -∞ < sup(𝐴, ℝ^*, < ))
15	13, 14	syl 17	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 supcsup 9434 ℝcr 11108 +∞cpnf 11246 -∞cmnf 11247 ℝ^*cxr 11248 < clt 11249

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448

This theorem is referenced by: supxrre1 13312 ovolunlem1a 25375 suplesup 44603

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