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Theorem xrltnsym 8868

Description: Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)

**Assertion**
Ref	Expression
xrltnsym	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

**Proof of Theorem xrltnsym**
Step	Hyp	Ref	Expression
1		elxr 8850	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 8850	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltnsym 7197	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4		rexr 7164	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
5		pnfnlt 8862	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → ¬ +∞ < 𝐴)
6	4, 5	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
7	6	adantr 270	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8		breq1 3788	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
9	8	adantl 271	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
10	7, 9	mtbird 630	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
11	10	a1d 22	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
12		nltmnf 8863	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
13	4, 12	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
14	13	adantr 270	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15		breq2 3789	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
16	15	adantl 271	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
17	14, 16	mtbird 630	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
18	17	pm2.21d 581	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
19	3, 11, 18	3jaodan 1237	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20		pnfnlt 8862	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
21	20	adantl 271	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
22		breq1 3788	. . . . . . 7 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
23	22	adantr 270	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
24	21, 23	mtbird 630	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ 𝐴 < 𝐵)
25	24	pm2.21d 581	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
26	2, 25	sylan2br 282	. . 3 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27		rexr 7164	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
28		nltmnf 8863	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
29	27, 28	syl 14	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
30	29	adantl 271	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31		breq2 3789	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐵 < 𝐴 ↔ 𝐵 < -∞))
32	31	adantr 270	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < -∞))
33	30, 32	mtbird 630	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < 𝐴)
34	33	a1d 22	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
35		mnfxr 8848	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
36		pnfnlt 8862	. . . . . . . 8 ⊢ (-∞ ∈ ℝ^* → ¬ +∞ < -∞)
37	35, 36	ax-mp 7	. . . . . . 7 ⊢ ¬ +∞ < -∞
38		breq12 3790	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ +∞ < -∞))
39	37, 38	mtbiri 632	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
40	39	ancoms 264	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
41	40	a1d 22	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
42		xrltnr 8855	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
43	35, 42	ax-mp 7	. . . . . 6 ⊢ ¬ -∞ < -∞
44		breq12 3790	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
45	43, 44	mtbiri 632	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
46	45	pm2.21d 581	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
47	34, 41, 46	3jaodan 1237	. . 3 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
48	19, 26, 47	3jaoian 1236	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
49	1, 2, 48	syl2anb 285	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ w3o 918 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 +∞cpnf 7150 -∞cmnf 7151 ℝ^*cxr 7152 < clt 7153

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-lttrn 7090

This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158

This theorem is referenced by: xrltnsym2 8869 xrltle 8873

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