Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rint0 | Structured version Visualization version 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 4839 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅) | |
2 | 1 | ineq2d 4103 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅)) |
3 | int0 4850 | . . . 4 ⊢ ∩ ∅ = V | |
4 | 3 | ineq2i 4100 | . . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V) |
5 | inv1 4283 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2761 | . 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴 |
7 | 2, 6 | eqtrdi 2789 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3398 ∩ cin 3842 ∅c0 4211 ∩ cint 4836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rab 3062 df-v 3400 df-dif 3846 df-in 3850 df-ss 3860 df-nul 4212 df-int 4837 |
This theorem is referenced by: incexclem 15286 incexc 15287 mrerintcl 16973 ismred2 16979 txtube 22393 bj-mooreset 34916 bj-ismoored0 34920 bj-ismooredr2 34924 |
Copyright terms: Public domain | W3C validator |