Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > renemnf | Structured version Visualization version GIF version |
Description: No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
renemnf | ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfnre 9961 | . . . 4 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2885 | . . 3 ⊢ ¬ -∞ ∈ ℝ |
3 | eleq1 2676 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 316 | . 2 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2811 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ℝcr 9814 -∞cmnf 9951 |
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-resscn 9872 |
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-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 |
This theorem is referenced by: renemnfd 9970 renfdisj 9977 xrnemnf 11827 rexneg 11916 rexadd 11937 xaddnemnf 11941 xaddcom 11945 xaddid1 11946 xnegdi 11950 xpncan 11953 xleadd1a 11955 rexmul 11973 xadddilem 11996 xrs1mnd 19603 xrs10 19604 isxmet2d 21942 imasdsf1olem 21988 xaddeq0 28907 icorempt2 32375 infrpge 38508 infleinflem1 38527 |
Copyright terms: Public domain | W3C validator |