![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xrsup0 | Structured version Visualization version GIF version |
Description: The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
xrsup0 | ⊢ sup(∅, ℝ*, < ) = -∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4397 | . 2 ⊢ ∅ ⊆ ℝ* | |
2 | mnfxr 11302 | . 2 ⊢ -∞ ∈ ℝ* | |
3 | ral0 4513 | . 2 ⊢ ∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 | |
4 | rexr 11291 | . . . . 5 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*) | |
5 | nltmnf 13142 | . . . . 5 ⊢ (𝑦 ∈ ℝ* → ¬ 𝑦 < -∞) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑦 ∈ ℝ → ¬ 𝑦 < -∞) |
7 | 6 | pm2.21d 121 | . . 3 ⊢ (𝑦 ∈ ℝ → (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)) |
8 | 7 | rgen 3060 | . 2 ⊢ ∀𝑦 ∈ ℝ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) |
9 | supxr 13325 | . 2 ⊢ (((∅ ⊆ ℝ* ∧ -∞ ∈ ℝ*) ∧ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) → sup(∅, ℝ*, < ) = -∞) | |
10 | 1, 2, 3, 8, 9 | mp4an 692 | 1 ⊢ sup(∅, ℝ*, < ) = -∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 ⊆ wss 3947 ∅c0 4323 class class class wbr 5148 supcsup 9464 ℝcr 11138 -∞cmnf 11277 ℝ*cxr 11278 < clt 11279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 |
This theorem is referenced by: mdegcl 26018 mdeg0 26019 suplesup 44721 supxrltinfxr 44831 supminfxr 44846 limsup0 45082 |
Copyright terms: Public domain | W3C validator |