nltpnft - Metamath Proof Explorer

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Theorem nltpnft 13124

Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

**Assertion**
Ref	Expression
nltpnft	⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

**Proof of Theorem nltpnft**
Step	Hyp	Ref	Expression
1		pnfxr 11228	. . . 4 ⊢ +∞ ∈ ℝ^*
2		xrltnr 13079	. . . 4 ⊢ (+∞ ∈ ℝ^* → ¬ +∞ < +∞)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ +∞ < +∞
4		breq1 5110	. . 3 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
5	3, 4	mtbiri 327	. 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞)
6		pnfge 13090	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
7		xrleloe 13104	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 ≤ +∞ ↔ (𝐴 < +∞ ∨ 𝐴 = +∞)))
8	1, 7	mpan2 691	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ≤ +∞ ↔ (𝐴 < +∞ ∨ 𝐴 = +∞)))
9	6, 8	mpbid 232	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < +∞ ∨ 𝐴 = +∞))
10	9	ord 864	. 2 ⊢ (𝐴 ∈ ℝ^* → (¬ 𝐴 < +∞ → 𝐴 = +∞))
11	5, 10	impbid2 226	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1540 ∈ wcel 2109 class class class wbr 5107 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-pre-lttri 11142 ax-pre-lttrn 11143

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214

This theorem is referenced by: xgepnf 13125 xrrebnd 13128 xlt2add 13220 supxrbnd1 13281 supxrbnd2 13282 supxrgtmnf 13289 supxrre2 13291 ioopnfsup 13826 icopnfsup 13827 xrsdsreclblem 21329 ovoliun 25406 ovolicopnf 25425 voliunlem3 25453 volsup 25457 itg2seq 25643 nmoreltpnf 30698 nmopreltpnf 31798 ismblfin 37655 supxrgere 45329 supxrgelem 45333 supxrge 45334 suplesup 45335 nepnfltpnf 45338 xrpnf 45481 sge0repnf 46384 sge0rpcpnf 46419 sge0rernmpt 46420

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