Theorem inssun 3373

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		pm3.1 754	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵))
2		eldifn 3256	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
3		eldifn 3256	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
4	2, 3	orim12i 759	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵))
5	1, 4	nsyl 628	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)))
6		elun 3274	. . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)))
7	5, 6	sylnibr 677	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
8		elin 3316	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9		vex 2738	. . . 4 ⊢ 𝑥 ∈ V
10		eldif 3136	. . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))))
11	9, 10	mpbiran 940	. . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
12	7, 8, 11	3imtr4i 201	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))))
13	12	ssriv 3157	1 ⊢ (𝐴 ∩ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 708   ∈ wcel 2146  Vcvv 2735   ∖ cdif 3124   ∪ cun 3125   ∩ cin 3126   ⊆ wss 3127

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140

This theorem is referenced by: (None)