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| Mirrors > Home > MPE Home > Th. List > mnfltxr | Structured version Visualization version GIF version | ||
| Description: Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.) |
| Ref | Expression |
|---|---|
| mnfltxr | ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mnflt 13165 | . 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
| 2 | mnfltpnf 13168 | . . 3 ⊢ -∞ < +∞ | |
| 3 | breq2 5147 | . . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
| 4 | 2, 3 | mpbiri 258 | . 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
| 5 | 1, 4 | jaoi 858 | 1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 +∞cpnf 11292 -∞cmnf 11293 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 |
| This theorem is referenced by: supxrgtmnf 13371 nmogtmnf 30789 nmopgtmnf 31887 |
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