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Theorem inssdif0im 3490
Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
Assertion
Ref Expression
inssdif0im ((𝐴𝐵) ⊆ 𝐶 → (𝐴 ∩ (𝐵𝐶)) = ∅)

Proof of Theorem inssdif0im
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3318 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21imbi1i 238 . . . . 5 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) → 𝑥𝐶))
3 imanim 688 . . . . 5 (((𝑥𝐴𝑥𝐵) → 𝑥𝐶) → ¬ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶))
42, 3sylbi 121 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) → ¬ ((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶))
5 eldif 3138 . . . . . 6 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
65anbi2i 457 . . . . 5 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
7 elin 3318 . . . . 5 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
8 anass 401 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
96, 7, 83bitr4ri 213 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ ¬ 𝑥𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶)))
104, 9sylnib 676 . . 3 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶)))
1110alimi 1455 . 2 (∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) → ∀𝑥 ¬ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶)))
12 dfss2 3144 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
13 eq0 3441 . 2 ((𝐴 ∩ (𝐵𝐶)) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶)))
1411, 12, 133imtr4i 201 1 ((𝐴𝐵) ⊆ 𝐶 → (𝐴 ∩ (𝐵𝐶)) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1351   = wceq 1353  wcel 2148  cdif 3126  cin 3128  wss 3129  c0 3422
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423
This theorem is referenced by:  disjdif  3495
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