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Theorem ssuni 4612
Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 26-Jul-2021.)
Assertion
Ref Expression
ssuni ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

Proof of Theorem ssuni
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elunii 4594 . . . . . 6 ((𝑦𝐵𝐵𝐶) → 𝑦 𝐶)
21expcom 450 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
32imim2d 57 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
43alimdv 1995 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
5 dfss2 3733 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
6 dfss2 3733 . . 3 (𝐴 𝐶 ↔ ∀𝑦(𝑦𝐴𝑦 𝐶))
74, 5, 63imtr4g 285 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 𝐶))
87impcom 445 1 ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630  wcel 2140  wss 3716   cuni 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-in 3723  df-ss 3730  df-uni 4590
This theorem is referenced by:  elssuni  4620  uniss2  4623  ssorduni  7152  filssufilg  21937  alexsubALTlem2  22074  utoptop  22260  locfinreflem  30238  setrec1  42967
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