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Theorem undif3 4031
 Description: An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.) (Proof shortened by JJ, 13-Jul-2021.)
Assertion
Ref Expression
undif3 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))

Proof of Theorem undif3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elun 3896 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 pm4.53 514 . . . . 5 (¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴) ↔ (¬ 𝑥𝐶𝑥𝐴))
3 eldif 3725 . . . . 5 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
42, 3xchnxbir 322 . . . 4 𝑥 ∈ (𝐶𝐴) ↔ (¬ 𝑥𝐶𝑥𝐴))
51, 4anbi12i 735 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
6 eldif 3725 . . 3 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
7 elun 3896 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
8 eldif 3725 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
98orbi2i 542 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
10 ordi 944 . . . . 5 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)))
11 orcom 401 . . . . . 6 ((𝑥𝐴 ∨ ¬ 𝑥𝐶) ↔ (¬ 𝑥𝐶𝑥𝐴))
1211anbi2i 732 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 ∨ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
1310, 12bitri 264 . . . 4 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
147, 9, 133bitri 286 . . 3 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
155, 6, 143bitr4ri 293 . 2 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
1615eqriv 2757 1 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ∖ cdif 3712   ∪ cun 3713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720 This theorem is referenced by:  undifabs  4189  llycmpkgen2  21555  hgt750lemb  31043
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