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Theorem uniss 3592
 Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniss (AB A B)

Proof of Theorem uniss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 2933 . . . . 5 (AB → (y Ay B))
21anim2d 320 . . . 4 (AB → ((x y y A) → (x y y B)))
32eximdv 1757 . . 3 (AB → (y(x y y A) → y(x y y B)))
4 eluni 3574 . . 3 (x Ay(x y y A))
5 eluni 3574 . . 3 (x By(x y y B))
63, 4, 53imtr4g 194 . 2 (AB → (x Ax B))
76ssrdv 2945 1 (AB A B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  ∪ cuni 3571 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572 This theorem is referenced by:  unissi  3594  unissd  3595  intssuni2m  3630  relfld  4789
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