![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > supxrnemnf | Structured version Visualization version GIF version |
Description: The supremum of a nonempty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
Ref | Expression |
---|---|
supxrnemnf | ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 11309 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → -∞ ∈ ℝ*) |
3 | supxrcl 13334 | . . 3 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
4 | 3 | 3ad2ant1 1130 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
5 | simp1 1133 | . . . 4 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ*) | |
6 | 5, 1 | jctir 519 | . . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → (𝐴 ⊆ ℝ* ∧ -∞ ∈ ℝ*)) |
7 | simpl 481 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ*) | |
8 | 7 | sselda 3982 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
9 | simpr 483 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
10 | simplr 767 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ¬ -∞ ∈ 𝐴) | |
11 | nelneq 2853 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) → ¬ 𝑥 = -∞) | |
12 | 9, 10, 11 | syl2anc 582 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = -∞) |
13 | ngtmnft 13185 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → (𝑥 = -∞ ↔ ¬ -∞ < 𝑥)) | |
14 | 13 | biimprd 247 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (¬ -∞ < 𝑥 → 𝑥 = -∞)) |
15 | 14 | con1d 145 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ* → (¬ 𝑥 = -∞ → -∞ < 𝑥)) |
16 | 8, 12, 15 | sylc 65 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → -∞ < 𝑥) |
17 | 16 | reximdva0 4355 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴) ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 -∞ < 𝑥) |
18 | 17 | 3impa 1107 | . . . 4 ⊢ ((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 -∞ < 𝑥) |
19 | 18 | 3com23 1123 | . . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → ∃𝑥 ∈ 𝐴 -∞ < 𝑥) |
20 | supxrlub 13344 | . . . 4 ⊢ ((𝐴 ⊆ ℝ* ∧ -∞ ∈ ℝ*) → (-∞ < sup(𝐴, ℝ*, < ) ↔ ∃𝑥 ∈ 𝐴 -∞ < 𝑥)) | |
21 | 20 | biimprd 247 | . . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ -∞ ∈ ℝ*) → (∃𝑥 ∈ 𝐴 -∞ < 𝑥 → -∞ < sup(𝐴, ℝ*, < ))) |
22 | 6, 19, 21 | sylc 65 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → -∞ < sup(𝐴, ℝ*, < )) |
23 | xrltne 13182 | . 2 ⊢ ((-∞ ∈ ℝ* ∧ sup(𝐴, ℝ*, < ) ∈ ℝ* ∧ -∞ < sup(𝐴, ℝ*, < )) → sup(𝐴, ℝ*, < ) ≠ -∞) | |
24 | 2, 4, 22, 23 | syl3anc 1368 | 1 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 supcsup 9471 -∞cmnf 11284 ℝ*cxr 11285 < clt 11286 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |