Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12512 | . 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
2 | mnfltpnf 12515 | . . 3 ⊢ -∞ < +∞ | |
3 | breq2 5062 | . . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
4 | 2, 3 | mpbiri 260 | . 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
5 | 1, 4 | jaoi 853 | 1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ℝcr 10530 +∞cpnf 10666 -∞cmnf 10667 < clt 10669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 |
This theorem is referenced by: supxrgtmnf 12716 nmogtmnf 28541 nmopgtmnf 29639 |
Copyright terms: Public domain | W3C validator |