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Theorem ssuni 3576
 Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
ssuni ((AB B 𝐶) → A 𝐶)

Proof of Theorem ssuni
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2083 . . . . . . 7 (x = B → (y xy B))
21imbi1d 220 . . . . . 6 (x = B → ((y xy 𝐶) ↔ (y By 𝐶)))
3 elunii 3559 . . . . . . 7 ((y x x 𝐶) → y 𝐶)
43expcom 109 . . . . . 6 (x 𝐶 → (y xy 𝐶))
52, 4vtoclga 2596 . . . . 5 (B 𝐶 → (y By 𝐶))
65imim2d 48 . . . 4 (B 𝐶 → ((y Ay B) → (y Ay 𝐶)))
76alimdv 1741 . . 3 (B 𝐶 → (y(y Ay B) → y(y Ay 𝐶)))
8 dfss2 2911 . . 3 (ABy(y Ay B))
9 dfss2 2911 . . 3 (A 𝐶y(y Ay 𝐶))
107, 8, 93imtr4g 194 . 2 (B 𝐶 → (ABA 𝐶))
1110impcom 116 1 ((AB B 𝐶) → A 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1226   = wceq 1228   ∈ wcel 1374   ⊆ wss 2894  ∪ cuni 3554 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555 This theorem is referenced by:  elssuni  3582  uniss2  3585  ssorduni  4163
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