| 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 5071 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) | |
| 2 | uniiun 5058 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥 | |
| 3 | 2 | ineq2i 4217 | . 2 ⊢ (𝐴 ∩ ∪ 𝐵) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) |
| 4 | 1, 3 | eqtr4i 2768 | 1 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∩ cin 3950 ∪ cuni 4907 ∪ ciun 4991 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-in 3958 df-uni 4908 df-iun 4993 |
| This theorem is referenced by: ssdifidllem 33484 ldgenpisyslem1 34164 |
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