ngtmnft - Metamath Proof Explorer

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Theorem ngtmnft 13218

Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)

**Assertion**
Ref	Expression
ngtmnft	⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))

**Proof of Theorem ngtmnft**
Step	Hyp	Ref	Expression
1		mnfxr 11290	. . . 4 ⊢ -∞ ∈ ℝ^*
2		xrltnr 13170	. . . 4 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ -∞ < -∞
4		breq2 5107	. . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞))
5	3, 4	mtbiri 330	. 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴)
6		mnfle 13186	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
7		xrleloe 13195	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
8	1, 7	mpan 703	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
9	6, 8	mpbid 235	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (-∞ < 𝐴 ∨ -∞ = 𝐴))
10	9	ord 878	. . 3 ⊢ (𝐴 ∈ ℝ^* → (¬ -∞ < 𝐴 → -∞ = 𝐴))
11		eqcom 2767	. . 3 ⊢ (-∞ = 𝐴 ↔ 𝐴 = -∞)
12	10, 11	imbitrdi 254	. 2 ⊢ (𝐴 ∈ ℝ^* → (¬ -∞ < 𝐴 → 𝐴 = -∞))
13	5, 12	impbid2 229	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 861 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 -∞cmnf 11265 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-pre-lttri 11198 ax-pre-lttrn 11199

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273

This theorem is used by: xlemnf 13219 xrrebnd 13220 ge0nemnf 13225 xlt2add 13312 xrsdsreclblem 21626 xblpnfps 24621 xblpnf 24622 supxrnemnf 33239 itg2addnclem 38420 supxrgelem 46167 supxrge 46168 nemnftgtmnft 46174 infxrbnd2 46198

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