Theorem ssundif 4399

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 1002	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
2		eldif 3876	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	2	imbi1i 353	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶))
4		elun 4063	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
5	4	imbi2i 339	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
6	1, 3, 5	3bitr4ri 307	. . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶))
7	6	albii 1827	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶))
8		dfss2 3886	. 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ 𝐶)))
9		dfss2 3886	. 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐶))
10	7, 8, 9	3bitr4i 306	1 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847  ∀wal 1541   ∈ wcel 2110   ∖ cdif 3863   ∪ cun 3864   ⊆ wss 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883

This theorem is referenced by:  difcom  4400  uneqdifeq  4404  ssunsn2  4740  elpwun  7554  soex  7699  ressuppssdif  7927  ssfi  8851  frfi  8916  cantnfp1lem3  9295  dfacfin7  10013  zornn0g  10119  fpwwe2lem12  10256  hashbclem  14016  incexclem  15400  ramub1lem1  16579  lpcls  22261  cmpcld  22299  alexsubALTlem3  22946  restmetu  23468  uniiccdif  24475  abelthlem2  25324  abelthlem3  25325  pmtrcnelor  31079  imadifss  35489  frege124d  41046