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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 4938 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆) | |
2 | 1 | ineq2d 4144 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆)) |
3 | 0iin 4989 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V | |
4 | 3 | ineq2i 4141 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V) |
5 | inv1 4326 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2767 | . 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴 |
7 | 2, 6 | eqtrdi 2796 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 Vcvv 3423 ∩ cin 3882 ∅c0 4254 ∩ ciin 4922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rab 3073 df-v 3425 df-dif 3886 df-in 3890 df-ss 3900 df-nul 4255 df-iin 4924 |
This theorem is referenced by: riinrab 5009 riiner 8496 mreriincl 17134 riinopn 21837 riincld 21973 fnemeet2 34327 pmapglb2N 37559 pmapglb2xN 37560 |
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