Theorem unssin 3443

Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)

Assertion

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

Proof of Theorem unssin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oranim 786	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldifn 3327	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
3		eldifn 3327	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
4	2, 3	anim12i 338	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	1, 4	nsyl 631	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵)))
6		elin 3387	. . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵)))
7	5, 6	sylnibr 681	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
8		elun 3345	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
9		vex 2802	. . . 4 ⊢ 𝑥 ∈ V
10		eldif 3206	. . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))))
11	9, 10	mpbiran 946	. . 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 3228	1 ⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 713   ∈ wcel 2200  Vcvv 2799   ∖ cdif 3194   ∪ cun 3195   ∩ cin 3196   ⊆ wss 3197

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by: (None)