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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 3924 | . 2 ⊢ ∅ ⊆ ℝ* | |
2 | mnfxr 9975 | . 2 ⊢ -∞ ∈ ℝ* | |
3 | ral0 4028 | . 2 ⊢ ∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 | |
4 | rexr 9964 | . . . . 5 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*) | |
5 | nltmnf 11839 | . . . . 5 ⊢ (𝑦 ∈ ℝ* → ¬ 𝑦 < -∞) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑦 ∈ ℝ → ¬ 𝑦 < -∞) |
7 | 6 | pm2.21d 117 | . . 3 ⊢ (𝑦 ∈ ℝ → (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)) |
8 | 7 | rgen 2906 | . 2 ⊢ ∀𝑦 ∈ ℝ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) |
9 | supxr 12015 | . 2 ⊢ (((∅ ⊆ ℝ* ∧ -∞ ∈ ℝ*) ∧ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) → sup(∅, ℝ*, < ) = -∞) | |
10 | 1, 2, 3, 8, 9 | mp4an 705 | 1 ⊢ sup(∅, ℝ*, < ) = -∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 supcsup 8229 ℝcr 9814 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 |
This theorem is referenced by: mdegcl 23633 mdeg0 23634 suplesup 38496 |
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