Theorem inssun 3362

Description: Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)

Assertion

Ref	Expression
inssun	⊢ (𝐴 ∩ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Proof of Theorem inssun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.1 744	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵))
2		eldifn 3245	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
3		eldifn 3245	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
4	2, 3	orim12i 749	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵))
5	1, 4	nsyl 618	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)))
6		elun 3263	. . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)))
7	5, 6	sylnibr 667	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
8		elin 3305	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9		vex 2729	. . . 4 ⊢ 𝑥 ∈ V
10		eldif 3125	. . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))))
11	9, 10	mpbiran 930	. . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
12	7, 8, 11	3imtr4i 200	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))))
13	12	ssriv 3146	1 ⊢ (𝐴 ∩ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 698   ∈ wcel 2136  Vcvv 2726   ∖ cdif 3113   ∪ cun 3114   ∩ cin 3115   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129

This theorem is referenced by: (None)