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Theorem unss 3111
Description: The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
Assertion
Ref Expression
unss ((A𝐶 B𝐶) ↔ (AB) ⊆ 𝐶)

Proof of Theorem unss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dfss2 2928 . 2 ((AB) ⊆ 𝐶x(x (AB) → x 𝐶))
2 19.26 1367 . . 3 (x((x Ax 𝐶) (x Bx 𝐶)) ↔ (x(x Ax 𝐶) x(x Bx 𝐶)))
3 elun 3078 . . . . . 6 (x (AB) ↔ (x A x B))
43imbi1i 227 . . . . 5 ((x (AB) → x 𝐶) ↔ ((x A x B) → x 𝐶))
5 jaob 630 . . . . 5 (((x A x B) → x 𝐶) ↔ ((x Ax 𝐶) (x Bx 𝐶)))
64, 5bitri 173 . . . 4 ((x (AB) → x 𝐶) ↔ ((x Ax 𝐶) (x Bx 𝐶)))
76albii 1356 . . 3 (x(x (AB) → x 𝐶) ↔ x((x Ax 𝐶) (x Bx 𝐶)))
8 dfss2 2928 . . . 4 (A𝐶x(x Ax 𝐶))
9 dfss2 2928 . . . 4 (B𝐶x(x Bx 𝐶))
108, 9anbi12i 433 . . 3 ((A𝐶 B𝐶) ↔ (x(x Ax 𝐶) x(x Bx 𝐶)))
112, 7, 103bitr4i 201 . 2 (x(x (AB) → x 𝐶) ↔ (A𝐶 B𝐶))
121, 11bitr2i 174 1 ((A𝐶 B𝐶) ↔ (AB) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628  wal 1240   wcel 1390  cun 2909  wss 2911
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-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by:  unssi  3112  unssd  3113  unssad  3114  unssbd  3115  nsspssun  3164  uneqin  3182  undifss  3297  prss  3511  prssg  3512  tpss  3520  pwundifss  4013  ordsucss  4196  elnn  4271  eqrelrel  4384  xpsspw  4393  relun  4397  relcoi2  4791  dfer2  6043  bdeqsuc  9316
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