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Theorem undif1 3625
 Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3622). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
undif1 ((A B) ∪ B) = (AB)

Proof of Theorem undif1
StepHypRef Expression
1 undir 3504 . 2 ((A ∩ (V B)) ∪ B) = ((AB) ∩ ((V B) ∪ B))
2 invdif 3496 . . 3 (A ∩ (V B)) = (A B)
32uneq1i 3414 . 2 ((A ∩ (V B)) ∪ B) = ((A B) ∪ B)
4 uncom 3408 . . . . 5 ((V B) ∪ B) = (B ∪ (V B))
5 undifv 3624 . . . . 5 (B ∪ (V B)) = V
64, 5eqtri 2373 . . . 4 ((V B) ∪ B) = V
76ineq2i 3454 . . 3 ((AB) ∩ ((V B) ∪ B)) = ((AB) ∩ V)
8 inv1 3577 . . 3 ((AB) ∩ V) = (AB)
97, 8eqtri 2373 . 2 ((AB) ∩ ((V B) ∪ B)) = (AB)
101, 3, 93eqtr3i 2381 1 ((A B) ∪ B) = (AB)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  undif2  3626  nnsucelrlem4  4427  ssfin  4470  sfinltfin  4535
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