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Mirrors > Home > ILE Home > Th. List > mnfltpnf | 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 2040 | . . . 4 ⊢ -∞ = -∞ | |
2 | eqid 2040 | . . . 4 ⊢ +∞ = +∞ | |
3 | olc 632 | . . . 4 ⊢ ((-∞ = -∞ ∧ +∞ = +∞) → (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞))) | |
4 | 1, 2, 3 | mp2an 402 | . . 3 ⊢ (((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) |
5 | 4 | orci 650 | . 2 ⊢ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))) |
6 | mnfxr 8694 | . . 3 ⊢ -∞ ∈ ℝ* | |
7 | pnfxr 8692 | . . 3 ⊢ +∞ ∈ ℝ* | |
8 | ltxr 8695 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ))))) | |
9 | 6, 7, 8 | mp2an 402 | . 2 ⊢ (-∞ < +∞ ↔ ((((-∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ -∞ <ℝ +∞) ∨ (-∞ = -∞ ∧ +∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ +∞ = +∞) ∨ (-∞ = -∞ ∧ +∞ ∈ ℝ)))) |
10 | 5, 9 | mpbir 134 | 1 ⊢ -∞ < +∞ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∨ wo 629 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 ℝcr 6888 <ℝ cltrr 6893 +∞cpnf 7057 -∞cmnf 7058 ℝ*cxr 7059 < clt 7060 |
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-pr 3944 ax-un 4170 ax-cnex 6975 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 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-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 |
This theorem is referenced by: mnfltxr 8707 xrlttr 8716 xrltso 8717 xrlttri3 8718 nltpnft 8730 ngtmnft 8731 xltnegi 8748 |
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