ngtmnft - Metamath Proof Explorer

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Theorem ngtmnft 13228

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 11347	. . . 4 ⊢ -∞ ∈ ℝ^*
2		xrltnr 13182	. . . 4 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ -∞ < -∞
4		breq2 5170	. . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞))
5	3, 4	mtbiri 327	. 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴)
6		mnfle 13197	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
7		xrleloe 13206	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
8	1, 7	mpan 689	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
9	6, 8	mpbid 232	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (-∞ < 𝐴 ∨ -∞ = 𝐴))
10	9	ord 863	. . 3 ⊢ (𝐴 ∈ ℝ^* → (¬ -∞ < 𝐴 → -∞ = 𝐴))
11		eqcom 2747	. . 3 ⊢ (-∞ = 𝐴 ↔ 𝐴 = -∞)
12	10, 11	imbitrdi 251	. 2 ⊢ (𝐴 ∈ ℝ^* → (¬ -∞ < 𝐴 → 𝐴 = -∞))
13	5, 12	impbid2 226	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 846 = wceq 1537 ∈ wcel 2108 class class class wbr 5166 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-pre-lttri 11258 ax-pre-lttrn 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330

This theorem is referenced by: xlemnf 13229 xrrebnd 13230 ge0nemnf 13235 xlt2add 13322 xrsdsreclblem 21453 xblpnfps 24426 xblpnf 24427 supxrnemnf 32775 itg2addnclem 37631 supxrgelem 45252 supxrge 45253 nemnftgtmnft 45259 infxrbnd2 45284

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