Theorem unssin 3203

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 840	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldifn 3095	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
3		eldifn 3095	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
4	2, 3	anim12i 331	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	1, 4	nsyl 590	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵)))
6		elin 3155	. . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ 𝑥 ∈ (V ∖ 𝐵)))
7	5, 6	sylnibr 634	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
8		elun 3113	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
9		vex 2604	. . . 4 ⊢ 𝑥 ∈ V
10		eldif 2982	. . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))))
11	9, 10	mpbiran 881	. . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
12	7, 8, 11	3imtr4i 199	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))))
13	12	ssriv 3003	1 ⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))

Syntax hints:  ¬ wn 3   ∧ wa 102   ∨ wo 661   ∈ wcel 1433  Vcvv 2601   ∖ cdif 2970   ∪ cun 2971   ∩ cin 2972   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by: (None)