Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  uniss GIF version

Theorem uniss 3601
 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 2939 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
21anim2d 320 . . . 4 (𝐴𝐵 → ((𝑥𝑦𝑦𝐴) → (𝑥𝑦𝑦𝐵)))
32eximdv 1760 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝑦𝑦𝐴) → ∃𝑦(𝑥𝑦𝑦𝐵)))
4 eluni 3583 . . 3 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
5 eluni 3583 . . 3 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
63, 4, 53imtr4g 194 . 2 (𝐴𝐵 → (𝑥 𝐴𝑥 𝐵))
76ssrdv 2951 1 (𝐴𝐵 𝐴 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ wcel 1393   ⊆ wss 2917  ∪ cuni 3580 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581 This theorem is referenced by:  unissi  3603  unissd  3604  intssuni2m  3639  relfld  4846
 Copyright terms: Public domain W3C validator