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Theorem ssuni 4647
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 4628 . . . . . 6 ((𝑦𝐵𝐵𝐶) → 𝑦 𝐶)
21expcom 400 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
32imim2d 57 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
43alimdv 2007 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
5 dfss2 3780 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
6 dfss2 3780 . . 3 (𝐴 𝐶 ↔ ∀𝑦(𝑦𝐴𝑦 𝐶))
74, 5, 63imtr4g 287 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 𝐶))
87impcom 396 1 ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635  wcel 2155  wss 3763   cuni 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-v 3389  df-in 3770  df-ss 3777  df-uni 4624
This theorem is referenced by:  elssuni  4654  uniss2  4657  ssorduni  7209  filssufilg  21922  alexsubALTlem2  22059  utoptop  22245  locfinreflem  30226  setrec1  43000
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