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Mirrors > Home > MPE Home > Th. List > undif2 | Structured version Visualization version GIF version |
Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4379). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.) |
Ref | Expression |
---|---|
undif2 | ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 4080 | . 2 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ 𝐴) | |
2 | undif1 4382 | . 2 ⊢ ((𝐵 ∖ 𝐴) ∪ 𝐴) = (𝐵 ∪ 𝐴) | |
3 | uncom 4080 | . 2 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
4 | 1, 2, 3 | 3eqtri 2825 | 1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∖ cdif 3878 ∪ cun 3879 |
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-or 845 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-un 3886 df-in 3888 df-ss 3898 df-nul 4244 |
This theorem is referenced by: undif 4388 dfif5 4441 funiunfv 6985 difex2 7462 undom 8588 sucdom2 8610 domss2 8660 unfi 8769 marypha1lem 8881 kmlem11 9571 hashun2 13740 hashun3 13741 cvgcmpce 15165 dprd2da 19157 dpjcntz 19167 dpjdisj 19168 dpjlsm 19169 dpjidcl 19173 ablfac1eu 19188 dfconn2 22024 2ndcdisj2 22062 fixufil 22527 fin1aufil 22537 xrge0gsumle 23438 unmbl 24141 volsup 24160 mbfss 24250 itg2cnlem2 24366 iblss2 24409 amgm 25576 wilthlem2 25654 ftalem3 25660 rpvmasum2 26096 esumpad 31424 srcmpltd 32456 noetalem3 33332 noetalem4 33333 imadifss 35032 elrfi 39635 meaunle 43103 |
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