nemnftgtmnft - Mathbox for Glauco Siliprandi

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Theorem nemnftgtmnft 45334

Description: An extended real that is not minus infinity, is larger than minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Assertion**
Ref	Expression
nemnftgtmnft	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)

**Proof of Theorem nemnftgtmnft**
Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞)
2	1	neneqd 2930	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → ¬ 𝐴 = -∞)
3		ngtmnft 13068	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
5	2, 4	mtbid 324	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → ¬ ¬ -∞ < 𝐴)
6		notnotb 315	. 2 ⊢ (-∞ < 𝐴 ↔ ¬ ¬ -∞ < 𝐴)
7	5, 6	sylibr 234	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5092 -∞cmnf 11147 ℝ^*cxr 11148 < clt 11149

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 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-pre-lttri 11083 ax-pre-lttrn 11084

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 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155

This theorem is referenced by: xrlexaddrp 45342

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