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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uniin2 | Structured version Visualization version GIF version |
Description: Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.) |
Ref | Expression |
---|---|
uniin2 | ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunin2 5075 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) | |
2 | uniiun 5062 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥 | |
3 | 2 | ineq2i 4224 | . 2 ⊢ (𝐴 ∩ ∪ 𝐵) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) |
4 | 1, 3 | eqtr4i 2765 | 1 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∩ cin 3961 ∪ cuni 4911 ∪ ciun 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-rab 3433 df-v 3479 df-in 3969 df-uni 4912 df-iun 4997 |
This theorem is referenced by: ssdifidllem 33463 ldgenpisyslem1 34143 |
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