difico - Mathbox for Thierry Arnoux

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Theorem difico 32713

Description: The difference between two closed-below, open-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 13-Oct-2017.)

**Assertion**
Ref	Expression
difico	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵))

**Proof of Theorem difico**
Step	Hyp	Ref	Expression
1		icodisj 13444	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅)
2		undif4 4433	. . . 4 ⊢ (((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶)))
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶)))
4	3	adantr 480	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶)))
5		difid 4342	. . . . 5 ⊢ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶)) = ∅
6	5	uneq2i 4131	. . . 4 ⊢ ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = ((𝐴[,)𝐵) ∪ ∅)
7		un0 4360	. . . 4 ⊢ ((𝐴[,)𝐵) ∪ ∅) = (𝐴[,)𝐵)
8	6, 7	eqtri 2753	. . 3 ⊢ ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (𝐴[,)𝐵)
9	8	a1i 11	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (𝐴[,)𝐵))
10		icoun 13443	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴[,)𝐶))
11	10	difeq1d 4091	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶)) = ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)))
12	4, 9, 11	3eqtr3rd 2774	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∖ cdif 3914 ∪ cun 3915 ∩ cin 3916 ∅c0 4299 class class class wbr 5110 (class class class)co 7390 ℝ^*cxr 11214 ≤ cle 11216 [,)cico 13315

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-pre-lttri 11149 ax-pre-lttrn 11150

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-ico 13319

This theorem is referenced by: sxbrsigalem2 34284

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