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Mirrors > Home > MPE Home > Th. List > nltpnft | Structured version Visualization version GIF version |
Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.) |
Ref | Expression |
---|---|
nltpnft | ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 10684 | . . . 4 ⊢ +∞ ∈ ℝ* | |
2 | xrltnr 12504 | . . . 4 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ +∞ < +∞ |
4 | breq1 5061 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
5 | 3, 4 | mtbiri 328 | . 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
6 | pnfge 12515 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
7 | xrleloe 12527 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ≤ +∞ ↔ (𝐴 < +∞ ∨ 𝐴 = +∞))) | |
8 | 1, 7 | mpan2 687 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ +∞ ↔ (𝐴 < +∞ ∨ 𝐴 = +∞))) |
9 | 6, 8 | mpbid 233 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 < +∞ ∨ 𝐴 = +∞)) |
10 | 9 | ord 858 | . 2 ⊢ (𝐴 ∈ ℝ* → (¬ 𝐴 < +∞ → 𝐴 = +∞)) |
11 | 5, 10 | impbid2 227 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: xgepnf 12548 xrrebnd 12551 xlt2add 12643 supxrbnd1 12704 supxrbnd2 12705 supxrgtmnf 12712 supxrre2 12714 ioopnfsup 13222 icopnfsup 13223 xrsdsreclblem 20521 ovoliun 24035 ovolicopnf 24054 voliunlem3 24082 volsup 24086 itg2seq 24272 nmoreltpnf 28474 nmopreltpnf 29574 ismblfin 34815 supxrgere 41481 supxrgelem 41485 supxrge 41486 suplesup 41487 nepnfltpnf 41490 xrpnf 41642 sge0repnf 42549 sge0rpcpnf 42584 sge0rernmpt 42585 |
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