ngtmnft - Metamath Proof Explorer

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Theorem ngtmnft 12829

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 10963	. . . 4 ⊢ -∞ ∈ ℝ^*
2		xrltnr 12784	. . . 4 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
3	1, 2	ax-mp 5	. . 3 ⊢ ¬ -∞ < -∞
4		breq2 5074	. . 3 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞))
5	3, 4	mtbiri 326	. 2 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴)
6		mnfle 12799	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
7		xrleloe 12807	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
8	1, 7	mpan 686	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
9	6, 8	mpbid 231	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (-∞ < 𝐴 ∨ -∞ = 𝐴))
10	9	ord 860	. . 3 ⊢ (𝐴 ∈ ℝ^* → (¬ -∞ < 𝐴 → -∞ = 𝐴))
11		eqcom 2745	. . 3 ⊢ (-∞ = 𝐴 ↔ 𝐴 = -∞)
12	10, 11	syl6ib 250	. 2 ⊢ (𝐴 ∈ ℝ^* → (¬ -∞ < 𝐴 → 𝐴 = -∞))
13	5, 12	impbid2 225	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 843 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 -∞cmnf 10938 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-pre-lttri 10876 ax-pre-lttrn 10877

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946

This theorem is referenced by: xlemnf 12830 xrrebnd 12831 ge0nemnf 12836 xlt2add 12923 xrsdsreclblem 20556 xblpnfps 23456 xblpnf 23457 supxrnemnf 30993 itg2addnclem 35755 supxrgelem 42766 supxrge 42767 nemnftgtmnft 42773 infxrbnd2 42798

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