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Mirrors > Home > MPE Home > Th. List > xrnemnf | Structured version Visualization version GIF version |
Description: An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xrnemnf | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.61 1000 | . 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞)) | |
2 | elxr 13038 | . . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
3 | df-3or 1089 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) | |
4 | 2, 3 | bitri 275 | . . 3 ⊢ (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) |
5 | df-ne 2945 | . . 3 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞) | |
6 | 4, 5 | anbi12i 628 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞)) |
7 | renemnf 11205 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
8 | pnfnemnf 11211 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
9 | neeq1 3007 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
10 | 8, 9 | mpbiri 258 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
11 | 7, 10 | jaoi 856 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → 𝐴 ≠ -∞) |
12 | 11 | neneqd 2949 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → ¬ 𝐴 = -∞) |
13 | 12 | pm4.71i 561 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞)) |
14 | 1, 6, 13 | 3bitr4i 303 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 |
This theorem is referenced by: xaddnemnf 13156 xaddass 13169 xlesubadd 13183 xrge0nre 13371 xblss2ps 23757 xblss2 23758 nmoix 24096 nmoleub 24098 blcvx 24164 xrge0tsms 24200 metdstri 24217 nmoleub2lem 24480 xrge0tsmsd 31902 esumcvgre 32693 icorempo 35825 xrnmnfpnf 43300 xrred 43606 |
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