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Theorem ssundifim 3521
Description: A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
ssundifim (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)

Proof of Theorem ssundifim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm5.6r 928 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
2 elun 3291 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32imbi2i 226 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 eldif 3153 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54imbi1i 238 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
61, 3, 53imtr4i 201 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) → (𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
76alimi 1466 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
8 dfss2 3159 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
9 dfss2 3159 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
107, 8, 93imtr4i 201 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  wal 1362  wcel 2160  cdif 3141  cun 3142  wss 3144
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157
This theorem is referenced by: (None)
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