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Theorem ssuni 3846
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 2253 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
21imbi1d 231 . . . . . 6 (𝑥 = 𝐵 → ((𝑦𝑥𝑦 𝐶) ↔ (𝑦𝐵𝑦 𝐶)))
3 elunii 3829 . . . . . . 7 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
43expcom 116 . . . . . 6 (𝑥𝐶 → (𝑦𝑥𝑦 𝐶))
52, 4vtoclga 2818 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
65imim2d 54 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
76alimdv 1890 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
8 dfss2 3159 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3159 . . 3 (𝐴 𝐶 ↔ ∀𝑦(𝑦𝐴𝑦 𝐶))
107, 8, 93imtr4g 205 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 𝐶))
1110impcom 125 1 ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1362   = wceq 1364  wcel 2160  wss 3144   cuni 3824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825
This theorem is referenced by:  elssuni  3852  uniss2  3855  ssorduni  4504
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