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Theorem ssundifim 3450
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 913 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
2 elun 3221 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32imbi2i 225 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 eldif 3084 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54imbi1i 237 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
61, 3, 53imtr4i 200 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) → (𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
76alimi 1432 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
8 dfss2 3090 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
9 dfss2 3090 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
107, 8, 93imtr4i 200 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698  wal 1330  wcel 1481  cdif 3072  cun 3073  wss 3075
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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088
This theorem is referenced by: (None)
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