MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  undif4 Structured version   Visualization version   GIF version

Theorem undif4 4012
Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undif4 ((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))

Proof of Theorem undif4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm2.621 424 . . . . . . 7 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) → ¬ 𝑥𝐶))
2 olc 399 . . . . . . 7 𝑥𝐶 → (𝑥𝐴 ∨ ¬ 𝑥𝐶))
31, 2impbid1 215 . . . . . 6 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴 ∨ ¬ 𝑥𝐶) ↔ ¬ 𝑥𝐶))
43anbi2d 739 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐶) → (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶)))
5 eldif 3569 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
65orbi2i 541 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
7 ordi 907 . . . . . 6 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
86, 7bitri 264 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
9 elun 3736 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
109anbi1i 730 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶))
114, 8, 103bitr4g 303 . . . 4 ((𝑥𝐴 → ¬ 𝑥𝐶) → ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶)))
12 elun 3736 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
13 eldif 3569 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥𝐶))
1411, 12, 133bitr4g 303 . . 3 ((𝑥𝐴 → ¬ 𝑥𝐶) → (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1514alimi 1736 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶) → ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
16 disj1 3996 . 2 ((𝐴𝐶) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐶))
17 dfcleq 2615 . 2 ((𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ 𝐶)))
1815, 16, 173imtr4i 281 1 ((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1987  cdif 3556  cun 3557  cin 3558  c0 3896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-nul 3897
This theorem is referenced by:  phplem1  8090  infdifsn  8505  difico  29407  caratheodorylem1  40068
  Copyright terms: Public domain W3C validator