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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uniin1 | 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 |
---|---|
uniin1 | ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝐴 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunin1 4805 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝑥 ∩ 𝐵) | |
2 | uniiun 4793 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | ineq1i 4037 | . 2 ⊢ (∪ 𝐴 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝑥 ∩ 𝐵) |
4 | 1, 3 | eqtr4i 2852 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∩ cin 3797 ∪ cuni 4658 ∪ ciun 4740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-v 3416 df-in 3805 df-ss 3812 df-uni 4659 df-iun 4742 |
This theorem is referenced by: (None) |
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