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Mirrors > Home > ILE Home > Th. List > rint0 | GIF version |
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 3862 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅) | |
2 | 1 | ineq2d 3351 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅)) |
3 | int0 3873 | . . . 4 ⊢ ∩ ∅ = V | |
4 | 3 | ineq2i 3348 | . . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V) |
5 | inv1 3474 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2210 | . 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴 |
7 | 2, 6 | eqtrdi 2238 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 Vcvv 2752 ∩ cin 3143 ∅c0 3437 ∩ cint 3859 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 df-int 3860 |
This theorem is referenced by: (None) |
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