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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 4898 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆) | |
2 | 1 | ineq2d 4139 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆)) |
3 | 0iin 4950 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V | |
4 | 3 | ineq2i 4136 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V) |
5 | inv1 4302 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2821 | . 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴 |
7 | 2, 6 | eqtrdi 2849 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 Vcvv 3441 ∩ cin 3880 ∅c0 4243 ∩ ciin 4882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-iin 4884 |
This theorem is referenced by: riinrab 4969 riiner 8353 mreriincl 16861 riinopn 21513 riincld 21649 fnemeet2 33828 pmapglb2N 37067 pmapglb2xN 37068 |
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