Theorem iccss2 12244

Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
iccss2	⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))

Proof of Theorem iccss2

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-icc 12182	. . . . . 6 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
2	1	elixx3g 12188	. . . . 5 ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
3	2	simplbi 476	. . . 4 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*))
4	3	adantr 481	. . 3 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*))
5	4	simp1d 1073	. 2 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ*)
6	4	simp2d 1074	. 2 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
7	2	simprbi 480	. . . 4 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
8	7	adantr 481	. . 3 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
9	8	simpld 475	. 2 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
10	1	elixx3g 12188	. . . . 5 ⊢ (𝐷 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ (𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵)))
11	10	simprbi 480	. . . 4 ⊢ (𝐷 ∈ (𝐴[,]𝐵) → (𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵))
12	11	simprd 479	. . 3 ⊢ (𝐷 ∈ (𝐴[,]𝐵) → 𝐷 ≤ 𝐵)
13	12	adantl 482	. 2 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐷 ≤ 𝐵)
14		xrletr 11989	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝑤) → 𝐴 ≤ 𝑤))
15		xrletr 11989	. . 3 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵) → 𝑤 ≤ 𝐵))
16	1, 1, 14, 15	ixxss12 12195	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
17	5, 6, 9, 13, 16	syl22anc 1327	1 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990   ⊆ wss 3574   class class class wbr 4653  (class class class)co 6650  ℝ^*cxr 10073   ≤ cle 10075  [,]cicc 12178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-icc 12182

This theorem is referenced by:  ordtresticc  21027  iccconn  22633  icccvx  22749  oprpiece1res1  22750  oprpiece1res2  22751  pcoass  22824  dvlip  23756  c1liplem1  23759  dvgt0lem1  23765  ftc2ditglem  23808  ttgcontlem1  25765  unitssxrge0  29946  xrge0iifhmeo  29982