nemnftgtmnft - Mathbox for Glauco Siliprandi

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Theorem nemnftgtmnft 45858

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 487	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞)
2	1	neneqd 2952	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → ¬ 𝐴 = -∞)
3		ngtmnft 13155	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
4	3	adantr 483	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
5	2, 4	mtbid 326	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → ¬ ¬ -∞ < 𝐴)
6		notnotb 317	. 2 ⊢ (-∞ < 𝐴 ↔ ¬ ¬ -∞ < 𝐴)
7	5, 6	sylibr 236	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1550 ∈ wcel 2132 ≠ wne 2947 class class class wbr 5090 -∞cmnf 11200 ℝ^*cxr 11201 < clt 11202

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-po 5544 df-so 5545 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-er 8662 df-en 8913 df-dom 8914 df-sdom 8915 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208

This theorem is referenced by: xrlexaddrp 45866

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