Theorem ssundif 4441

Description: A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)

Assertion

Ref	Expression
ssundif	⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶)

Proof of Theorem ssundif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.6 1003	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
2		eldif 3915	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	2	imbi1i 349	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶))
4		elun 4106	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
5	4	imbi2i 336	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
6	1, 3, 5	3bitr4ri 304	. . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶))
7	6	albii 1819	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶))
8		df-ss 3922	. 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)))
9		df-ss 3922	. 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶))
10	7, 8, 9	3bitr4i 303	1 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1538   ∈ wcel 2109   ∖ cdif 3902   ∪ cun 3903   ⊆ wss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922

This theorem is referenced by:  difcom  4442  uneqdifeq  4446  ssunsn2  4781  f1imadifssran  6572  elpwun  7709  soex  7861  ressuppssdif  8125  ssfi  9097  frfi  9190  cantnfp1lem3  9595  dfacfin7  10312  zornn0g  10418  fpwwe2lem12  10555  hashbclem  14377  incexclem  15761  ramub1lem1  16956  lpcls  23267  cmpcld  23305  alexsubALTlem3  23952  restmetu  24474  uniiccdif  25495  abelthlem2  26358  abelthlem3  26359  pmtrcnelor  33046  imadifss  37577  frege124d  43737