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Theorem xrltnsym 9729

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 9712	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 9712	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltnsym 7984	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4		rexr 7944	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
5		pnfnlt 9723	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → ¬ +∞ < 𝐴)
6	4, 5	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
7	6	adantr 274	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8		breq1 3985	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
9	8	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
10	7, 9	mtbird 663	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
11	10	a1d 22	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
12		nltmnf 9724	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
13	4, 12	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
14	13	adantr 274	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15		breq2 3986	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
16	15	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
17	14, 16	mtbird 663	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
18	17	pm2.21d 609	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
19	3, 11, 18	3jaodan 1296	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20		pnfnlt 9723	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
21	20	adantl 275	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
22		breq1 3985	. . . . . . 7 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
23	22	adantr 274	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
24	21, 23	mtbird 663	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ 𝐴 < 𝐵)
25	24	pm2.21d 609	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
26	2, 25	sylan2br 286	. . 3 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27		rexr 7944	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
28		nltmnf 9724	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
29	27, 28	syl 14	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
30	29	adantl 275	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31		breq2 3986	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐵 < 𝐴 ↔ 𝐵 < -∞))
32	31	adantr 274	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < -∞))
33	30, 32	mtbird 663	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < 𝐴)
34	33	a1d 22	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
35		mnfxr 7955	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
36		pnfnlt 9723	. . . . . . . 8 ⊢ (-∞ ∈ ℝ^* → ¬ +∞ < -∞)
37	35, 36	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < -∞
38		breq12 3987	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ +∞ < -∞))
39	37, 38	mtbiri 665	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
40	39	ancoms 266	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
41	40	a1d 22	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
42		xrltnr 9715	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
43	35, 42	ax-mp 5	. . . . . 6 ⊢ ¬ -∞ < -∞
44		breq12 3987	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
45	43, 44	mtbiri 665	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
46	45	pm2.21d 609	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
47	34, 41, 46	3jaodan 1296	. . 3 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
48	19, 26, 47	3jaoian 1295	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
49	1, 2, 48	syl2anb 289	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 967 = wceq 1343 ∈ wcel 2136 class class class wbr 3982 ℝcr 7752 +∞cpnf 7930 -∞cmnf 7931 ℝ^*cxr 7932 < clt 7933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-pre-ltirr 7865 ax-pre-lttrn 7867

This theorem depends on definitions: df-bi 116 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-xp 4610 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938

This theorem is referenced by: xrltnsym2 9730 xrltle 9734

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