nemnftgtmnft - Mathbox for Glauco Siliprandi

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Theorem nemnftgtmnft 44639

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 2940	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → ¬ 𝐴 = -∞)
3		ngtmnft 13163	. . . 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 233	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 -∞cmnf 11262 ℝ^*cxr 11263 < clt 11264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-pre-lttri 11198 ax-pre-lttrn 11199

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270

This theorem is referenced by: xrlexaddrp 44647

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