Theorem xrltnsym 12176

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 12156	. 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 12156	. 2 ⊢ (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltnsym 10338	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4		rexr 10288	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
5		pnfnlt 12168	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
7	6	adantr 466	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8		breq1 4790	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
9	8	adantl 467	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
10	7, 9	mtbird 314	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
11	10	a1d 25	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
12		nltmnf 12169	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
13	4, 12	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
14	13	adantr 466	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15		breq2 4791	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
16	15	adantl 467	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
17	14, 16	mtbird 314	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
18	17	pm2.21d 119	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
19	3, 11, 18	3jaodan 1542	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20		pnfnlt 12168	. . . . . . 7 ⊢ (𝐵 ∈ ℝ* → ¬ +∞ < 𝐵)
21	20	adantl 467	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ +∞ < 𝐵)
22		breq1 4790	. . . . . . 7 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
23	22	adantr 466	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
24	21, 23	mtbird 314	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐴 < 𝐵)
25	24	pm2.21d 119	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
26	2, 25	sylan2br 576	. . 3 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27		rexr 10288	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
28		nltmnf 12169	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ* → ¬ 𝐵 < -∞)
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
30	29	adantl 467	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31		breq2 4791	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐵 < 𝐴 ↔ 𝐵 < -∞))
32	31	adantr 466	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < -∞))
33	30, 32	mtbird 314	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < 𝐴)
34	33	a1d 25	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
35		mnfxr 10299	. . . . . . . 8 ⊢ -∞ ∈ ℝ*
36		pnfnlt 12168	. . . . . . . 8 ⊢ (-∞ ∈ ℝ* → ¬ +∞ < -∞)
37	35, 36	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < -∞
38		breq12 4792	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ +∞ < -∞))
39	37, 38	mtbiri 316	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
40	39	ancoms 455	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
41	40	a1d 25	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
42		xrltnr 12159	. . . . . . 7 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞)
43	35, 42	ax-mp 5	. . . . . 6 ⊢ ¬ -∞ < -∞
44		breq12 4792	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
45	43, 44	mtbiri 316	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
46	45	pm2.21d 119	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
47	34, 41, 46	3jaodan 1542	. . 3 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
48	19, 26, 47	3jaoian 1541	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
49	1, 2, 48	syl2anb 579	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∨ w3o 1070   = wceq 1631   ∈ wcel 2145   class class class wbr 4787  ℝcr 10138  +∞cpnf 10274  -∞cmnf 10275  ℝ^*cxr 10276   < clt 10277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-pre-lttri 10213  ax-pre-lttrn 10214

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-po 5171  df-so 5172  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282

This theorem is referenced by:  xrltnsym2  12177  xrlttri  12178  xmullem2  12301  sgnp  14039  iccpartnel  41903