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Theorem ssuni 3681
Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
ssuni ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

Proof of Theorem ssuni
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2152 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
21imbi1d 230 . . . . . 6 (𝑥 = 𝐵 → ((𝑦𝑥𝑦 𝐶) ↔ (𝑦𝐵𝑦 𝐶)))
3 elunii 3664 . . . . . . 7 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
43expcom 115 . . . . . 6 (𝑥𝐶 → (𝑦𝑥𝑦 𝐶))
52, 4vtoclga 2686 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
65imim2d 54 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
76alimdv 1808 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
8 dfss2 3015 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3015 . . 3 (𝐴 𝐶 ↔ ∀𝑦(𝑦𝐴𝑦 𝐶))
107, 8, 93imtr4g 204 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 𝐶))
1110impcom 124 1 ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1288   = wceq 1290  wcel 1439  wss 3000   cuni 3659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-uni 3660
This theorem is referenced by:  elssuni  3687  uniss2  3690  ssorduni  4317
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