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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 4882 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅) | |
2 | 1 | ineq2d 4192 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅)) |
3 | int0 4893 | . . . 4 ⊢ ∩ ∅ = V | |
4 | 3 | ineq2i 4189 | . . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V) |
5 | inv1 4351 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2847 | . 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴 |
7 | 2, 6 | syl6eq 2875 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Vcvv 3497 ∩ cin 3938 ∅c0 4294 ∩ cint 4879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 df-nul 4295 df-int 4880 |
This theorem is referenced by: incexclem 15194 incexc 15195 mrerintcl 16871 ismred2 16877 txtube 22251 bj-mooreset 34398 bj-ismoored0 34402 bj-ismooredr2 34406 |
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