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| Description: Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| riin0 | ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iineq1 5008 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆) | |
| 2 | 1 | ineq2d 4219 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆)) | 
| 3 | 0iin 5063 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V | |
| 4 | 3 | ineq2i 4216 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V) | 
| 5 | inv1 4397 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
| 6 | 4, 5 | eqtri 2764 | . 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴 | 
| 7 | 2, 6 | eqtrdi 2792 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 Vcvv 3479 ∩ cin 3949 ∅c0 4332 ∩ ciin 4991 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-in 3957 df-ss 3967 df-nul 4333 df-iin 4993 | 
| This theorem is referenced by: riinrab 5083 riiner 8831 mreriincl 17642 riinopn 22915 riincld 23053 fnemeet2 36369 pmapglb2N 39774 pmapglb2xN 39775 | 
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