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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 4927 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆) | |
2 | 1 | ineq2d 4186 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆)) |
3 | 0iin 4978 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V | |
4 | 3 | ineq2i 4183 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V) |
5 | inv1 4345 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2841 | . 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴 |
7 | 2, 6 | syl6eq 2869 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 Vcvv 3492 ∩ cin 3932 ∅c0 4288 ∩ ciin 4911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-iin 4913 |
This theorem is referenced by: riinrab 4997 riiner 8359 mreriincl 16857 riinopn 21444 riincld 21580 fnemeet2 33612 pmapglb2N 36787 pmapglb2xN 36788 |
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