ngtmnft - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ngtmnft		Structured version Visualization version GIF version

Theorem ngtmnft 13204

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 11277	. . . 4 ⊢ -∞ ∈ ℝ^*
2		xrltnr 13156	. . . 4 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ -∞ < -∞
4		breq2 5115	. . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞))
5	3, 4	mtbiri 330	. 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴)
6		mnfle 13172	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
7		xrleloe 13181	. . . . . 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 2772	. . 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 2146 class class class wbr 5111 -∞cmnf 11252 ℝ^*cxr 11253 < clt 11254 ≤ cle 11255

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-pre-lttri 11185 ax-pre-lttrn 11186

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260

This theorem is used by: xlemnf 13205 xrrebnd 13206 ge0nemnf 13211 xlt2add 13298 xrsdsreclblem 21593 xblpnfps 24583 xblpnf 24584 supxrnemnf 33159 itg2addnclem 38355 supxrgelem 46086 supxrge 46087 nemnftgtmnft 46093 infxrbnd2 46117

W3C validator