Theorem inssdif 3339

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 3227	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ (V ∖ 𝐵))
2	1	anim2i 340	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
3		elin 3286	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
4		eldif 3107	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
5	2, 3, 4	3imtr4i 200	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
6	5	ssriv 3128	1 ⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 103   ∈ wcel 2125  Vcvv 2709   ∖ cdif 3095   ∩ cin 3097   ⊆ wss 3098

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111

This theorem is referenced by:  difdif2ss  3360