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Mirrors > Home > MPE Home > Th. List > riin0 | Structured version Visualization version 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 4729 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆) | |
2 | 1 | ineq2d 4016 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆)) |
3 | 0iin 4772 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V | |
4 | 3 | ineq2i 4013 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V) |
5 | inv1 4170 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2825 | . 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴 |
7 | 2, 6 | syl6eq 2853 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 Vcvv 3389 ∩ cin 3772 ∅c0 4119 ∩ ciin 4715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2781 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ral 3098 df-v 3391 df-dif 3776 df-in 3780 df-ss 3787 df-nul 4120 df-iin 4717 |
This theorem is referenced by: riinrab 4790 riiner 8062 mreriincl 16577 riinopn 21045 riincld 21181 fnemeet2 32878 pmapglb2N 35796 pmapglb2xN 35797 |
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