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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 13043 | . 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
2 | mnfltpnf 13046 | . . 3 ⊢ -∞ < +∞ | |
3 | breq2 5109 | . . 3 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
4 | 2, 3 | mpbiri 257 | . 2 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
5 | 1, 4 | jaoi 855 | 1 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 ℝcr 11049 +∞cpnf 11185 -∞cmnf 11186 < clt 11188 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-xp 5639 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 |
This theorem is referenced by: supxrgtmnf 13247 nmogtmnf 29659 nmopgtmnf 30757 |
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