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Mirrors > Home > MPE Home > Th. List > inundif | Structured version Visualization version GIF version |
Description: The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
inundif | ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4053 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | eldif 3835 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | orbi12i 898 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) |
4 | pm4.42 1034 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) | |
5 | 3, 4 | bitr4i 270 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ 𝑥 ∈ 𝐴) |
6 | 5 | uneqri 4012 | 1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2048 ∖ cdif 3822 ∪ cun 3823 ∩ cin 3824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 |
This theorem is referenced by: iunxdif3 4877 resasplit 6371 fresaun 6372 fresaunres2 6373 ixpfi2 8609 hashun3 13551 prmreclem2 16099 mvdco 18324 sylow2a 18495 ablfac1eu 18935 basdif0 21255 neitr 21482 cmpfi 21710 ptbasfi 21883 ptcnplem 21923 fin1aufil 22234 ismbl2 23821 volinun 23840 voliunlem2 23845 mbfeqalem2 23936 itg2cnlem2 24056 dvres2lem 24201 indifundif 30049 imadifxp 30107 ofpreima2 30163 partfun 30173 resf1o 30207 gsummptres 30485 indsumin 30882 measun 31072 measunl 31077 inelcarsg 31171 carsgclctun 31181 sibfof 31200 probdif 31281 hgt750lemd 31528 mthmpps 32289 clcnvlem 39291 radcnvrat 40006 sumnnodd 41288 ovolsplit 41650 omelesplit 42177 ovnsplit 42307 |
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