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Theorem ssundifim 3334
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 847 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
2 elun 3112 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32imbi2i 219 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 eldif 2955 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54imbi1i 231 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
61, 3, 53imtr4i 194 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) → (𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
76alimi 1360 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
8 dfss2 2962 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
9 dfss2 2962 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
107, 8, 93imtr4i 194 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  wal 1257  wcel 1409  cdif 2942  cun 2943  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959
This theorem is referenced by: (None)
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