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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > supxrmnf2 | Structured version Visualization version GIF version |
Description: Removing minus infinity from a set does not affect its supremum. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
supxrmnf2 | ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdifss 3939 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ∖ {-∞}) ⊆ ℝ*) | |
2 | supxrmnf 12396 | . . . . 5 ⊢ ((𝐴 ∖ {-∞}) ⊆ ℝ* → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < )) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < )) |
4 | 3 | adantr 473 | . . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴) → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < )) |
5 | difsnid 4529 | . . . . 5 ⊢ (-∞ ∈ 𝐴 → ((𝐴 ∖ {-∞}) ∪ {-∞}) = 𝐴) | |
6 | 5 | supeq1d 8594 | . . . 4 ⊢ (-∞ ∈ 𝐴 → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
7 | 6 | adantl 474 | . . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴) → sup(((𝐴 ∖ {-∞}) ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
8 | 4, 7 | eqtr3d 2835 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
9 | difsn 4517 | . . . 4 ⊢ (¬ -∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) = 𝐴) | |
10 | 9 | supeq1d 8594 | . . 3 ⊢ (¬ -∞ ∈ 𝐴 → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
11 | 10 | adantl 474 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
12 | 8, 11 | pm2.61dan 848 | 1 ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∖ cdif 3766 ∪ cun 3767 ⊆ wss 3769 {csn 4368 supcsup 8588 -∞cmnf 10361 ℝ*cxr 10362 < clt 10363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 |
This theorem is referenced by: supminfxr2 40442 |
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