New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > undif2 | GIF version |
Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3622). 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 3408 | . 2 ⊢ (A ∪ (B ∖ A)) = ((B ∖ A) ∪ A) | |
2 | undif1 3625 | . 2 ⊢ ((B ∖ A) ∪ A) = (B ∪ A) | |
3 | uncom 3408 | . 2 ⊢ (B ∪ A) = (A ∪ B) | |
4 | 1, 2, 3 | 3eqtri 2377 | 1 ⊢ (A ∪ (B ∖ A)) = (A ∪ B) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∖ cdif 3206 ∪ cun 3207 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: undif 3630 dfif5 3674 dflec2 6210 |
Copyright terms: Public domain | W3C validator |