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Theorem nltpnft 12827

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 10960	. . . 4 ⊢ +∞ ∈ ℝ^*
2		xrltnr 12784	. . . 4 ⊢ (+∞ ∈ ℝ^* → ¬ +∞ < +∞)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ +∞ < +∞
4		breq1 5073	. . 3 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
5	3, 4	mtbiri 326	. 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞)
6		pnfge 12795	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
7		xrleloe 12807	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 ≤ +∞ ↔ (𝐴 < +∞ ∨ 𝐴 = +∞)))
8	1, 7	mpan2 687	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ≤ +∞ ↔ (𝐴 < +∞ ∨ 𝐴 = +∞)))
9	6, 8	mpbid 231	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < +∞ ∨ 𝐴 = +∞))
10	9	ord 860	. 2 ⊢ (𝐴 ∈ ℝ^* → (¬ 𝐴 < +∞ → 𝐴 = +∞))
11	5, 10	impbid2 225	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 843 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-pre-lttri 10876 ax-pre-lttrn 10877

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946

This theorem is referenced by: xgepnf 12828 xrrebnd 12831 xlt2add 12923 supxrbnd1 12984 supxrbnd2 12985 supxrgtmnf 12992 supxrre2 12994 ioopnfsup 13512 icopnfsup 13513 xrsdsreclblem 20556 ovoliun 24574 ovolicopnf 24593 voliunlem3 24621 volsup 24625 itg2seq 24812 nmoreltpnf 29032 nmopreltpnf 30132 ismblfin 35745 supxrgere 42762 supxrgelem 42766 supxrge 42767 suplesup 42768 nepnfltpnf 42771 xrpnf 42916 sge0repnf 43814 sge0rpcpnf 43849 sge0rernmpt 43850

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