Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssuni Structured version   Visualization version   GIF version

Theorem ssuni 4852
 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 4835 . . . . . 6 ((𝑦𝐵𝐵𝐶) → 𝑦 𝐶)
21expcom 416 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
32imim2d 57 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
43alimdv 1910 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
5 dfss2 3953 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
6 dfss2 3953 . . 3 (𝐴 𝐶 ↔ ∀𝑦(𝑦𝐴𝑦 𝐶))
74, 5, 63imtr4g 298 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 𝐶))
87impcom 410 1 ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1528   ∈ wcel 2107   ⊆ wss 3934  ∪ cuni 4830 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-in 3941  df-ss 3950  df-uni 4831 This theorem is referenced by:  elssuni  4859  uniss2  4862  ssorduni  7492  filssufilg  22511  alexsubALTlem2  22648  utoptop  22835  locfinreflem  31092  bj-sselpwuni  34330  setrec1  44774
 Copyright terms: Public domain W3C validator