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 4295 | . 2 ⊢ ∅ ⊆ ℝ* | |
2 | mnfxr 10788 | . 2 ⊢ -∞ ∈ ℝ* | |
3 | ral0 4409 | . 2 ⊢ ∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 | |
4 | rexr 10777 | . . . . 5 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*) | |
5 | nltmnf 12619 | . . . . 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 3064 | . 2 ⊢ ∀𝑦 ∈ ℝ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) |
9 | supxr 12801 | . 2 ⊢ (((∅ ⊆ ℝ* ∧ -∞ ∈ ℝ*) ∧ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) → sup(∅, ℝ*, < ) = -∞) | |
10 | 1, 2, 3, 8, 9 | mp4an 693 | 1 ⊢ sup(∅, ℝ*, < ) = -∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∃wrex 3055 ⊆ wss 3853 ∅c0 4221 class class class wbr 5040 supcsup 8989 ℝcr 10626 -∞cmnf 10763 ℝ*cxr 10764 < clt 10765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 |
This theorem is referenced by: mdegcl 24834 mdeg0 24835 suplesup 42456 supxrltinfxr 42568 supminfxr 42584 limsup0 42817 |
Copyright terms: Public domain | W3C validator |