nepnfltpnf - Mathbox for Glauco Siliprandi

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Theorem nepnfltpnf 45247

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 2951	. . 3 ⊢ (𝜑 → ¬ 𝐴 = +∞)
3		nepnfltpnf.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		nltpnft 13220	. . . 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 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 +∞cpnf 11315 ℝ^*cxr 11317 < clt 11318

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7764 ax-cnex 11234 ax-resscn 11235 ax-pre-lttri 11252 ax-pre-lttrn 11253

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5701 df-rel 5702 df-cnv 5703 df-co 5704 df-dm 5705 df-rn 5706 df-res 5707 df-ima 5708 df-iota 6520 df-fun 6570 df-fn 6571 df-f 6572 df-f1 6573 df-fo 6574 df-f1o 6575 df-fv 6576 df-er 8757 df-en 8998 df-dom 8999 df-sdom 9000 df-pnf 11320 df-mnf 11321 df-xr 11322 df-ltxr 11323 df-le 11324

This theorem is referenced by: infrpge 45256