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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 2733 | . . . 4 ⊢ -∞ = -∞ | |
2 | eqid 2733 | . . . 4 ⊢ +∞ = +∞ | |
3 | olc 867 | . . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))) | |
4 | 1, 2, 3 | mp2an 691 | . . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) |
5 | 4 | orci 864 | . 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))) |
6 | mnfxr 11266 | . . 3 ⊢ -∞ ∈ ℝ* | |
7 | pnfxr 11263 | . . 3 ⊢ +∞ ∈ ℝ* | |
8 | ltxr 13090 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))))) | |
9 | 6, 7, 8 | mp2an 691 | . 2 ⊢ (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))) |
10 | 5, 9 | mpbir 230 | 1 ⊢ -∞ < +∞ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5146 ℝcr 11104 <ℝ cltrr 11109 +∞cpnf 11240 -∞cmnf 11241 ℝ*cxr 11242 < clt 11243 |
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 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-xp 5680 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 |
This theorem is referenced by: mnfltxr 13102 xrlttri 13113 xrlttr 13114 xltnegi 13190 supxrltinfxr 44093 liminflelimsupcex 44447 |
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