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Mirrors > Home > ILE Home > Th. List > riin0 | GIF version |
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 3900 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆) | |
2 | 1 | ineq2d 3336 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆)) |
3 | 0iin 3944 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V | |
4 | 3 | ineq2i 3333 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V) |
5 | inv1 3459 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2198 | . 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴 |
7 | 2, 6 | eqtrdi 2226 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 Vcvv 2737 ∩ cin 3128 ∅c0 3422 ∩ ciin 3887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 df-iin 3889 |
This theorem is referenced by: (None) |
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