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Theorem ssundifim 3347
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 870 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
2 elun 3125 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32imbi2i 224 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 eldif 2993 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54imbi1i 236 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
61, 3, 53imtr4i 199 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) → (𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
76alimi 1385 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
8 dfss2 2999 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
9 dfss2 2999 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
107, 8, 93imtr4i 199 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  wal 1283  wcel 1434  cdif 2981  cun 2982  wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997
This theorem is referenced by: (None)
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