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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 11833 | . 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
2 | mnfltpnf 11836 | . . 3 ⊢ -∞ < +∞ | |
3 | breq2 4587 | . . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
4 | 2, 3 | mpbiri 247 | . 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
5 | 1, 4 | jaoi 393 | 1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 +∞cpnf 9950 -∞cmnf 9951 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 |
This theorem is referenced by: supxrgtmnf 12031 nmogtmnf 27009 nmopgtmnf 28111 |
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