nepnfltpnf - Mathbox for Glauco Siliprandi

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Theorem nepnfltpnf 45451

Description: An extended real that is not +∞ is less than +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
nepnfltpnf.1	⊢ (𝜑 → 𝐴 ≠ +∞)
nepnfltpnf.2	⊢ (𝜑 → 𝐴 ∈ ℝ^*)

**Assertion**
Ref	Expression
nepnfltpnf	⊢ (𝜑 → 𝐴 < +∞)

**Proof of Theorem nepnfltpnf**
Step	Hyp	Ref	Expression
1		nepnfltpnf.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ +∞)
2	1	neneqd 2933	. . 3 ⊢ (𝜑 → ¬ 𝐴 = +∞)
3		nepnfltpnf.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		nltpnft 13063	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
6	2, 5	mtbid 324	. 2 ⊢ (𝜑 → ¬ ¬ 𝐴 < +∞)
7		notnotb 315	. 2 ⊢ (𝐴 < +∞ ↔ ¬ ¬ 𝐴 < +∞)
8	6, 7	sylibr 234	1 ⊢ (𝜑 → 𝐴 < +∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 class class class wbr 5089 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-pre-lttri 11080 ax-pre-lttrn 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152

This theorem is referenced by: infrpge 45460