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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 12506 | . 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
2 | mnfltpnf 12509 | . . 3 ⊢ -∞ < +∞ | |
3 | breq2 5034 | . . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
4 | 2, 3 | mpbiri 261 | . 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
5 | 1, 4 | jaoi 854 | 1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ℝcr 10525 +∞cpnf 10661 -∞cmnf 10662 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 |
This theorem is referenced by: supxrgtmnf 12710 nmogtmnf 28553 nmopgtmnf 29651 |
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