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Mirrors > Home > MPE Home > Th. List > elxr | Structured version Visualization version GIF version |
Description: Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
elxr | ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xr 11194 | . . 3 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ 𝐴 ∈ (ℝ ∪ {+∞, -∞})) |
3 | elun 4109 | . 2 ⊢ (𝐴 ∈ (ℝ ∪ {+∞, -∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 ∈ {+∞, -∞})) | |
4 | pnfex 11209 | . . . . 5 ⊢ +∞ ∈ V | |
5 | mnfxr 11213 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
6 | 5 | elexi 3465 | . . . . 5 ⊢ -∞ ∈ V |
7 | 4, 6 | elpr2 4612 | . . . 4 ⊢ (𝐴 ∈ {+∞, -∞} ↔ (𝐴 = +∞ ∨ 𝐴 = -∞)) |
8 | 7 | orbi2i 912 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {+∞, -∞}) ↔ (𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞))) |
9 | 3orass 1091 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞))) | |
10 | 8, 9 | bitr4i 278 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {+∞, -∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
11 | 2, 3, 10 | 3bitri 297 | 1 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 {cpr 4589 ℝcr 11051 +∞cpnf 11187 -∞cmnf 11188 ℝ*cxr 11189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-pow 5321 ax-un 7673 ax-cnex 11108 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3448 df-un 3916 df-in 3918 df-ss 3928 df-pw 4563 df-sn 4588 df-pr 4590 df-uni 4867 df-pnf 11192 df-mnf 11193 df-xr 11194 |
This theorem is referenced by: xrnemnf 13039 xrnepnf 13040 xrltnr 13041 xrltnsym 13057 xrlttri 13059 xrlttr 13060 xrrebnd 13088 qbtwnxr 13120 xnegcl 13133 xnegneg 13134 xltnegi 13136 xaddf 13144 xnegid 13158 xaddcom 13160 xaddid1 13161 xnegdi 13168 xleadd1a 13173 xlt2add 13180 xsubge0 13181 xmullem 13184 xmulid1 13199 xmulgt0 13203 xmulasslem3 13206 xlemul1a 13208 xadddilem 13214 xadddi2 13217 xrsupsslem 13227 xrinfmsslem 13228 xrub 13232 reltxrnmnf 13262 isxmet2d 23683 blssioo 24161 ioombl1 24929 ismbf2d 25007 itg2seq 25110 xaddeq0 31661 iooelexlt 35836 relowlssretop 35837 iccpartiltu 45621 iccpartigtl 45622 |
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