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Theorem xrltnsym 13051

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 13030	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13030	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltnsym 11231	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4		rexr 11178	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
5		pnfnlt 13042	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → ¬ +∞ < 𝐴)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8		breq1 5101	. . . . . . 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 13043	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
13	4, 12	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
14	13	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15		breq2 5102	. . . . . . 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 1433	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20		pnfnlt 13042	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
21	20	adantl 481	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
22		breq1 5101	. . . . . . 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 595	. . 3 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27		rexr 11178	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
28		nltmnf 13043	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
30	29	adantl 481	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31		breq2 5102	. . . . . . 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 11189	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
36		pnfnlt 13042	. . . . . . . 8 ⊢ (-∞ ∈ ℝ^* → ¬ +∞ < -∞)
37	35, 36	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < -∞
38		breq12 5103	. . . . . . 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 13033	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
43	35, 42	ax-mp 5	. . . . . 6 ⊢ ¬ -∞ < -∞
44		breq12 5103	. . . . . 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 1433	. . 3 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
48	19, 26, 47	3jaoian 1432	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
49	1, 2, 48	syl2anb 598	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 1085 = wceq 1541 ∈ wcel 2113 class class class wbr 5098 ℝcr 11025 +∞cpnf 11163 -∞cmnf 11164 ℝ^*cxr 11165 < clt 11166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-pre-lttri 11100 ax-pre-lttrn 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171

This theorem is referenced by: xrltnsym2 13052 xrlttri 13053 xmullem2 13180 sgnp 15013 iccpartnel 47684

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