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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 4872 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅) | |
2 | 1 | ineq2d 4189 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅)) |
3 | int0 4883 | . . . 4 ⊢ ∩ ∅ = V | |
4 | 3 | ineq2i 4186 | . . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V) |
5 | inv1 4348 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2844 | . 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴 |
7 | 2, 6 | syl6eq 2872 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3495 ∩ cin 3935 ∅c0 4291 ∩ cint 4869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3497 df-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 df-int 4870 |
This theorem is referenced by: incexclem 15185 incexc 15186 mrerintcl 16862 ismred2 16868 txtube 22242 bj-mooreset 34388 bj-ismoored0 34392 bj-ismooredr2 34396 |
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