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Theorem nltpnft 13107

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 11190	. . . 4 ⊢ +∞ ∈ ℝ^*
2		xrltnr 13061	. . . 4 ⊢ (+∞ ∈ ℝ^* → ¬ +∞ < +∞)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ +∞ < +∞
4		breq1 5089	. . 3 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
5	3, 4	mtbiri 327	. 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞)
6		pnfge 13072	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
7		xrleloe 13086	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 ≤ +∞ ↔ (𝐴 < +∞ ∨ 𝐴 = +∞)))
8	1, 7	mpan2 692	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ≤ +∞ ↔ (𝐴 < +∞ ∨ 𝐴 = +∞)))
9	6, 8	mpbid 232	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < +∞ ∨ 𝐴 = +∞))
10	9	ord 865	. 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 848 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-pre-lttri 11103 ax-pre-lttrn 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176

This theorem is referenced by: xgepnf 13108 xrrebnd 13111 xlt2add 13203 supxrbnd1 13264 supxrbnd2 13265 supxrgtmnf 13272 supxrre2 13274 ioopnfsup 13814 icopnfsup 13815 xrsdsreclblem 21402 ovoliun 25482 ovolicopnf 25501 voliunlem3 25529 volsup 25533 itg2seq 25719 nmoreltpnf 30855 nmopreltpnf 31955 ismblfin 37996 supxrgere 45781 supxrgelem 45785 supxrge 45786 suplesup 45787 nepnfltpnf 45790 xrpnf 45931 sge0repnf 46832 sge0rpcpnf 46867 sge0rernmpt 46868

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