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Theorem ssuni 4425
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 4407 . . . . . 6 ((𝑦𝐵𝐵𝐶) → 𝑦 𝐶)
21expcom 451 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
32imim2d 57 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
43alimdv 1842 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
5 dfss2 3572 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
6 dfss2 3572 . . 3 (𝐴 𝐶 ↔ ∀𝑦(𝑦𝐴𝑦 𝐶))
74, 5, 63imtr4g 285 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 𝐶))
87impcom 446 1 ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wcel 1987  wss 3555   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-in 3562  df-ss 3569  df-uni 4403
This theorem is referenced by:  elssuni  4433  uniss2  4436  ssorduni  6932  filssufilg  21625  alexsubALTlem2  21762  utoptop  21948  locfinreflem  29686  setrec1  41728
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