ge0gtmnf - Metamath Proof Explorer

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Theorem ge0gtmnf 13175

Description: A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
ge0gtmnf	⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)

**Proof of Theorem ge0gtmnf**
Step	Hyp	Ref	Expression
1		mnflt0 13127	. 2 ⊢ -∞ < 0
2		mnfxr 11239	. . . 4 ⊢ -∞ ∈ ℝ^*
3		0xr 11229	. . . 4 ⊢ 0 ∈ ℝ^*
4		xrltletr 13159	. . . 4 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴))
5	2, 3, 4	mp3an12 1472	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴))
6	5	imp 410	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ (-∞ < 0 ∧ 0 ≤ 𝐴)) → -∞ < 𝐴)
7	1, 6	mpanr1 713	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2142 class class class wbr 5100 0cc0 11073 -∞cmnf 11214 ℝ^*cxr 11215 < clt 11216 ≤ cle 11217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-addrcl 11134 ax-rnegex 11144 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-po 5555 df-so 5556 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222

This theorem is referenced by: ge0nemnf 13176 xrrege0 13177 pcgcd1 16913 pnfnei 23277 isnghm3 24782 ovolunnul 25559 radcnvle 26480 psercnlem1 26485