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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 5071 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) | |
2 | uniiun 5058 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥 | |
3 | 2 | ineq2i 4207 | . 2 ⊢ (𝐴 ∩ ∪ 𝐵) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) |
4 | 1, 3 | eqtr4i 2757 | 1 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∩ cin 3945 ∪ cuni 4905 ∪ ciun 4993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rex 3061 df-rab 3420 df-v 3464 df-in 3953 df-uni 4906 df-iun 4995 |
This theorem is referenced by: ssdifidllem 33337 ldgenpisyslem1 34009 |
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