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Theorem ssundifim 3306
 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 836 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
2 elun 3084 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32imbi2i 215 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 eldif 2927 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54imbi1i 227 . . . 4 ((𝑥 ∈ (𝐴𝐵) → 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥𝐶))
61, 3, 53imtr4i 190 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) → (𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
76alimi 1344 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
8 dfss2 2934 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
9 dfss2 2934 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → 𝑥𝐶))
107, 8, 93imtr4i 190 1 (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 629  ∀wal 1241   ∈ wcel 1393   ∖ cdif 2914   ∪ cun 2915   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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