difico - Mathbox for Thierry Arnoux

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Theorem difico 33139

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 13509	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅)
2		undif4 4426	. . . 4 ⊢ (((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶)))
3	1, 2	syl 18	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = (((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) ∖ (𝐵[,)𝐶)))
4	3	adantr 485	. 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 4331	. . . . 5 ⊢ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶)) = ∅
6	5	uneq2i 4118	. . . 4 ⊢ ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = ((𝐴[,)𝐵) ∪ ∅)
7		un0 4350	. . . 4 ⊢ ((𝐴[,)𝐵) ∪ ∅) = (𝐴[,)𝐵)
8	6, 7	eqtri 2785	. . 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 13508	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴[,)𝐶))
11	10	difeq1d 4079	. 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 2806	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ∖ cdif 3901 ∪ cun 3902 ∩ cin 3903 ∅c0 4285 class class class wbr 5108 (class class class)co 7412 ℝ^*cxr 11248 ≤ cle 11250 [,)cico 13380

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-po 5568 df-so 5569 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-ico 13384

This theorem is used by: sxbrsigalem2 34685

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