Theorem inssdif 3408

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

Assertion

Ref	Expression
inssdif	⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))

Proof of Theorem inssdif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elndif 3296	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ (V ∖ 𝐵))
2	1	anim2i 342	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
3		elin 3355	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
4		eldif 3174	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
5	2, 3, 4	3imtr4i 201	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
6	5	ssriv 3196	1 ⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∈ wcel 2175  Vcvv 2771   ∖ cdif 3162   ∩ cin 3164   ⊆ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178

This theorem is referenced by:  difdif2ss  3429