Theorem nltle2tri 47761

Description: Negated extended trichotomy law for 'less than' and 'less than or equal to'. (Contributed by AV, 18-Jul-2020.)

Assertion

Ref	Expression
nltle2tri	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))

Proof of Theorem nltle2tri

Step	Hyp	Ref	Expression
1		xrltletr 13108	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶))
2		id 22	. . . . . . . . . 10 ⊢ (((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶))
3	2	impcom 407	. . . . . . . . 9 ⊢ (((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∧ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) → 𝐴 < 𝐶)
4		xrltnle 11212	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐴))
5	4	3adant2 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐴))
6	5	biimpd 229	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐶 → ¬ 𝐶 ≤ 𝐴))
7	6	imp 406	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 < 𝐶) → ¬ 𝐶 ≤ 𝐴)
8	7	olcd 875	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 < 𝐶) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴))
9	8	expcom 413	. . . . . . . . 9 ⊢ (𝐴 < 𝐶 → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴)))
10	3, 9	syl 17	. . . . . . . 8 ⊢ (((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∧ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴)))
11	10	ex 412	. . . . . . 7 ⊢ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → (((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴))))
12	11	com23 86	. . . . . 6 ⊢ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴))))
13	12	impd 410	. . . . 5 ⊢ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴)))
14		id 22	. . . . . . 7 ⊢ (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → ¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶))
15	14	orcd 874	. . . . . 6 ⊢ (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴))
16	15	a1d 25	. . . . 5 ⊢ (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴)))
17	13, 16	pm2.61i 182	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) → (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴))
18		df-3an 1089	. . . . . 6 ⊢ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∧ 𝐶 ≤ 𝐴))
19	18	notbii 320	. . . . 5 ⊢ (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ ¬ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∧ 𝐶 ≤ 𝐴))
20		ianor 984	. . . . 5 ⊢ (¬ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∧ 𝐶 ≤ 𝐴) ↔ (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴))
21	19, 20	bitri 275	. . . 4 ⊢ (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ ¬ 𝐶 ≤ 𝐴))
22	17, 21	sylibr 234	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) → ¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))
23	22	ex 412	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶) → ¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)))
24	1, 23	mpd 15	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ¬ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   ∈ wcel 2114   class class class wbr 5085  ℝ^*cxr 11178   < clt 11179   ≤ cle 11180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185

This theorem is referenced by:  icceuelpart  47896