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Theorem ssuni 3753
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 2201 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
21imbi1d 230 . . . . . 6 (𝑥 = 𝐵 → ((𝑦𝑥𝑦 𝐶) ↔ (𝑦𝐵𝑦 𝐶)))
3 elunii 3736 . . . . . . 7 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
43expcom 115 . . . . . 6 (𝑥𝐶 → (𝑦𝑥𝑦 𝐶))
52, 4vtoclga 2747 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
65imim2d 54 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
76alimdv 1851 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
8 dfss2 3081 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3081 . . 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 1329   = wceq 1331  wcel 1480  wss 3066   cuni 3731
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732
This theorem is referenced by:  elssuni  3759  uniss2  3762  ssorduni  4398
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