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Theorem xrltnsym 13199

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 13179	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13179	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltnsym 11388	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4		rexr 11336	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
5		pnfnlt 13191	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → ¬ +∞ < 𝐴)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8		breq1 5169	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
9	8	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
10	7, 9	mtbird 325	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
11	10	a1d 25	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
12		nltmnf 13192	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
13	4, 12	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
14	13	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15		breq2 5170	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
16	15	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
17	14, 16	mtbird 325	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
18	17	pm2.21d 121	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
19	3, 11, 18	3jaodan 1431	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20		pnfnlt 13191	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
21	20	adantl 481	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
22		breq1 5169	. . . . . . 7 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
23	22	adantr 480	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
24	21, 23	mtbird 325	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ 𝐴 < 𝐵)
25	24	pm2.21d 121	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
26	2, 25	sylan2br 594	. . 3 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27		rexr 11336	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
28		nltmnf 13192	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
30	29	adantl 481	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31		breq2 5170	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐵 < 𝐴 ↔ 𝐵 < -∞))
32	31	adantr 480	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < -∞))
33	30, 32	mtbird 325	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < 𝐴)
34	33	a1d 25	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
35		mnfxr 11347	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
36		pnfnlt 13191	. . . . . . . 8 ⊢ (-∞ ∈ ℝ^* → ¬ +∞ < -∞)
37	35, 36	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < -∞
38		breq12 5171	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ +∞ < -∞))
39	37, 38	mtbiri 327	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
40	39	ancoms 458	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
41	40	a1d 25	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
42		xrltnr 13182	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
43	35, 42	ax-mp 5	. . . . . 6 ⊢ ¬ -∞ < -∞
44		breq12 5171	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
45	43, 44	mtbiri 327	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
46	45	pm2.21d 121	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
47	34, 41, 46	3jaodan 1431	. . 3 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
48	19, 26, 47	3jaoian 1430	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
49	1, 2, 48	syl2anb 597	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1086 = wceq 1537 ∈ wcel 2108 class class class wbr 5166 ℝcr 11183 +∞cpnf 11321 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324

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

This theorem is referenced by: xrltnsym2 13200 xrlttri 13201 xmullem2 13327 sgnp 15139 iccpartnel 47312

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