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Mirrors > Home > MPE Home > Th. List > indifcom | Structured version Visualization version GIF version |
Description: Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.) |
Ref | Expression |
---|---|
indifcom | ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4128 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | difeq1i 4046 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐵 ∩ 𝐴) ∖ 𝐶) |
3 | indif2 4197 | . 2 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶) | |
4 | indif2 4197 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶) | |
5 | 2, 3, 4 | 3eqtr4i 2831 | 1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∖ cdif 3878 ∩ cin 3880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 |
This theorem is referenced by: ufprim 22514 cmmbl 24138 unmbl 24141 volinun 24150 limciun 24497 caragenuncllem 43151 |
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