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| Description: Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| rint0 | ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | inteq 4949 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅) | |
| 2 | 1 | ineq2d 4220 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅)) | 
| 3 | int0 4962 | . . . 4 ⊢ ∩ ∅ = V | |
| 4 | 3 | ineq2i 4217 | . . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V) | 
| 5 | inv1 4398 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
| 6 | 4, 5 | eqtri 2765 | . 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴 | 
| 7 | 2, 6 | eqtrdi 2793 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 Vcvv 3480 ∩ cin 3950 ∅c0 4333 ∩ cint 4946 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-int 4947 | 
| This theorem is referenced by: incexclem 15872 incexc 15873 mrerintcl 17640 ismred2 17646 txtube 23648 bj-mooreset 37103 bj-ismoored0 37107 bj-ismooredr2 37111 | 
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