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Mirrors > Home > MPE Home > Th. List > inundif | Structured version Visualization version Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3796 | . . . 4 | |
2 | eldif 3584 | . . . 4 | |
3 | 1, 2 | orbi12i 543 | . . 3 |
4 | pm4.42 1004 | . . 3 | |
5 | 3, 4 | bitr4i 267 | . 2 |
6 | 5 | uneqri 3755 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wo 383 wa 384 wceq 1483 wcel 1990 cdif 3571 cun 3572 cin 3573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 |
This theorem is referenced by: iunxdif3 4606 resasplit 6074 fresaun 6075 fresaunres2 6076 ixpfi2 8264 hashun3 13173 prmreclem2 15621 mvdco 17865 sylow2a 18034 ablfac1eu 18472 basdif0 20757 neitr 20984 cmpfi 21211 ptbasfi 21384 ptcnplem 21424 fin1aufil 21736 ismbl2 23295 volinun 23314 voliunlem2 23319 mbfeqalem 23409 itg2cnlem2 23529 dvres2lem 23674 indifundif 29356 imadifxp 29414 ofpreima2 29466 partfun 29475 resf1o 29505 gsummptres 29784 indsumin 30084 measun 30274 measunl 30279 inelcarsg 30373 carsgclctun 30383 sibfof 30402 probdif 30482 hgt750lemd 30726 mthmpps 31479 clcnvlem 37930 radcnvrat 38513 sumnnodd 39862 ovolsplit 40205 omelesplit 40732 ovnsplit 40862 |
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