difico - Mathbox for Thierry Arnoux

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Theorem difico 32535

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 13477	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅)
2		undif4 4462	. . . 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 4366	. . . . 5 ⊢ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶)) = ∅
6	5	uneq2i 4156	. . . 4 ⊢ ((𝐴[,)𝐵) ∪ ((𝐵[,)𝐶) ∖ (𝐵[,)𝐶))) = ((𝐴[,)𝐵) ∪ ∅)
7		un0 4386	. . . 4 ⊢ ((𝐴[,)𝐵) ∪ ∅) = (𝐴[,)𝐵)
8	6, 7	eqtri 2755	. . 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 13476	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴[,)𝐶))
11	10	difeq1d 4117	. 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 2776	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 1085 = wceq 1534 ∈ wcel 2099 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ∅c0 4318 class class class wbr 5142 (class class class)co 7414 ℝ^*cxr 11269 ≤ cle 11271 [,)cico 13350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-pre-lttri 11204 ax-pre-lttrn 11205

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-ico 13354

This theorem is referenced by: sxbrsigalem2 33842

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