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Mirrors > Home > MPE Home > Th. List > mnfltpnf | Structured version Visualization version GIF version |
Description: Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
mnfltpnf | ⊢ -∞ < +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . 4 ⊢ -∞ = -∞ | |
2 | eqid 2734 | . . . 4 ⊢ +∞ = +∞ | |
3 | olc 868 | . . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))) | |
4 | 1, 2, 3 | mp2an 692 | . . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) |
5 | 4 | orci 865 | . 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))) |
6 | mnfxr 11315 | . . 3 ⊢ -∞ ∈ ℝ* | |
7 | pnfxr 11312 | . . 3 ⊢ +∞ ∈ ℝ* | |
8 | ltxr 13154 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))))) | |
9 | 6, 7, 8 | mp2an 692 | . 2 ⊢ (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))) |
10 | 5, 9 | mpbir 231 | 1 ⊢ -∞ < +∞ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ℝcr 11151 <ℝ cltrr 11156 +∞cpnf 11289 -∞cmnf 11290 ℝ*cxr 11291 < clt 11292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 |
This theorem is referenced by: mnfltxr 13166 xrlttri 13177 xrlttr 13178 xltnegi 13254 supxrltinfxr 45398 liminflelimsupcex 45752 |
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