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Theorem undif 3630
 Description: Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
undif (A B ↔ (A ∪ (B A)) = B)

Proof of Theorem undif
StepHypRef Expression
1 ssequn1 3433 . 2 (A B ↔ (AB) = B)
2 undif2 3626 . . 3 (A ∪ (B A)) = (AB)
32eqeq1i 2360 . 2 ((A ∪ (B A)) = B ↔ (AB) = B)
41, 3bitr4i 243 1 (A B ↔ (A ∪ (B A)) = B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∖ cdif 3206   ∪ cun 3207   ⊆ wss 3257 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:  difsnid  3854
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