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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 3965 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | eldif 3959 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | orbi12i 914 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) |
4 | pm4.42 1053 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) | |
5 | 3, 4 | bitr4i 278 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ 𝑥 ∈ 𝐴) |
6 | 5 | uneqri 4152 | 1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 |
This theorem is referenced by: iunxdif3 5099 partfun 6698 resasplit 6762 fresaun 6763 fresaunres2 6764 ixpfi2 9350 hashun3 14344 prmreclem2 16850 mvdco 19313 sylow2a 19487 ablfac1eu 19943 basdif0 22456 neitr 22684 cmpfi 22912 ptbasfi 23085 ptcnplem 23125 fin1aufil 23436 ismbl2 25044 volinun 25063 voliunlem2 25068 mbfeqalem2 25159 itg2cnlem2 25280 dvres2lem 25427 indifundif 31762 imadifxp 31832 ofpreima2 31891 resf1o 31955 gsummptres 32204 tocyccntz 32303 indsumin 33020 measun 33209 measunl 33214 inelcarsg 33310 carsgclctun 33320 sibfof 33339 probdif 33419 hgt750lemd 33660 mthmpps 34573 clcnvlem 42374 radcnvrat 43073 sumnnodd 44346 ovolsplit 44704 omelesplit 45234 ovnsplit 45364 |
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