Theorem iccss2 13370

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 13305	. . . . . 6 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
2	1	elixx3g 13311	. . . . 5 ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
3	2	simplbi 496	. . . 4 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*))
4	3	adantr 480	. . 3 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*))
5	4	simp1d 1143	. 2 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ*)
6	4	simp2d 1144	. 2 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
7	2	simprbi 497	. . . 4 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
8	7	adantr 480	. . 3 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
9	8	simpld 494	. 2 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
10	1	elixx3g 13311	. . . . 5 ⊢ (𝐷 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ (𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵)))
11	10	simprbi 497	. . . 4 ⊢ (𝐷 ∈ (𝐴[,]𝐵) → (𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵))
12	11	simprd 495	. . 3 ⊢ (𝐷 ∈ (𝐴[,]𝐵) → 𝐷 ≤ 𝐵)
13	12	adantl 481	. 2 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐷 ≤ 𝐵)
14		xrletr 13109	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝑤) → 𝐴 ≤ 𝑤))
15		xrletr 13109	. . 3 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵) → 𝑤 ≤ 𝐵))
16	1, 1, 14, 15	ixxss12 13318	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
17	5, 6, 9, 13, 16	syl22anc 839	1 ⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114   ⊆ wss 3889   class class class wbr 5085  (class class class)co 7367  ℝ^*cxr 11178   ≤ cle 11180  [,]cicc 13301

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-iun 4935  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-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  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  df-icc 13305

This theorem is referenced by:  ordtresticc  23188  iccconn  24796  icccvx  24917  oprpiece1res1  24918  oprpiece1res2  24919  pcoass  24991  dvlip  25960  c1liplem1  25963  dvgt0lem1  25969  ftc2ditglem  26012  ttgcontlem1  28953  unitssxrge0  34044  xrge0iifhmeo  34080