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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 2731 | . . . 4 ⊢ -∞ = -∞ | |
2 | eqid 2731 | . . . 4 ⊢ +∞ = +∞ | |
3 | olc 865 | . . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))) | |
4 | 1, 2, 3 | mp2an 689 | . . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) |
5 | 4 | orci 862 | . 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))) |
6 | mnfxr 11276 | . . 3 ⊢ -∞ ∈ ℝ* | |
7 | pnfxr 11273 | . . 3 ⊢ +∞ ∈ ℝ* | |
8 | ltxr 13100 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))))) | |
9 | 6, 7, 8 | mp2an 689 | . 2 ⊢ (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))) |
10 | 5, 9 | mpbir 230 | 1 ⊢ -∞ < +∞ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 class class class wbr 5149 ℝcr 11112 <ℝ cltrr 11117 +∞cpnf 11250 -∞cmnf 11251 ℝ*cxr 11252 < clt 11253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 |
This theorem is referenced by: mnfltxr 13112 xrlttri 13123 xrlttr 13124 xltnegi 13200 supxrltinfxr 44459 liminflelimsupcex 44813 |
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