Theorem unssdif 3394

Description: Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)

Assertion

Ref	Expression
unssdif	⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Proof of Theorem unssdif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2763	. . . . . . . 8 ⊢ 𝑥 ∈ V
2		eldif 3162	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
3	1, 2	mpbiran 942	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
4	3	anbi1i 458	. . . . . 6 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		eldif 3162	. . . . . 6 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵))
6		ioran 753	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
7	4, 5, 6	3bitr4i 212	. . . . 5 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
8	7	biimpi 120	. . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) → ¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
9	8	con2i 628	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
10		elun 3300	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
11		eldif 3162	. . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)))
12	1, 11	mpbiran 942	. . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
13	9, 10, 12	3imtr4i 201	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
14	13	ssriv 3183	1 ⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 709   ∈ wcel 2164  Vcvv 2760   ∖ cdif 3150   ∪ cun 3151   ⊆ wss 3153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166

This theorem is referenced by: (None)