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Mirrors > Home > ILE Home > Th. List > elxr | 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 7064 | . . 3 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
2 | 1 | eleq2i 2104 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ 𝐴 ∈ (ℝ ∪ {+∞, -∞})) |
3 | elun 3084 | . 2 ⊢ (𝐴 ∈ (ℝ ∪ {+∞, -∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 ∈ {+∞, -∞})) | |
4 | pnfex 8693 | . . . . 5 ⊢ +∞ ∈ V | |
5 | mnfxr 8694 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
6 | 5 | elexi 2567 | . . . . 5 ⊢ -∞ ∈ V |
7 | 4, 6 | elpr2 3397 | . . . 4 ⊢ (𝐴 ∈ {+∞, -∞} ↔ (𝐴 = +∞ ∨ 𝐴 = -∞)) |
8 | 7 | orbi2i 679 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {+∞, -∞}) ↔ (𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞))) |
9 | 3orass 888 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞))) | |
10 | 8, 9 | bitr4i 176 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {+∞, -∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
11 | 2, 3, 10 | 3bitri 195 | 1 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∨ wo 629 ∨ w3o 884 = wceq 1243 ∈ wcel 1393 ∪ cun 2915 {cpr 3376 ℝcr 6888 +∞cpnf 7057 -∞cmnf 7058 ℝ*cxr 7059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-un 4170 ax-cnex 6975 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-pnf 7062 df-mnf 7063 df-xr 7064 |
This theorem is referenced by: xrnemnf 8699 xrnepnf 8700 xrltnr 8701 xrltnsym 8714 xrlttr 8716 xrltso 8717 xrlttri3 8718 nltpnft 8730 ngtmnft 8731 xrrebnd 8732 xnegcl 8745 xnegneg 8746 xltnegi 8748 |
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