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

Theorem iunss 3688
 Description: Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss ( x A B𝐶x A B𝐶)
Distinct variable group:   x,𝐶
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem iunss
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-iun 3649 . . 3 x A B = {yx A y B}
21sseq1i 2963 . 2 ( x A B𝐶 ↔ {yx A y B} ⊆ 𝐶)
3 abss 3003 . 2 ({yx A y B} ⊆ 𝐶y(x A y By 𝐶))
4 dfss2 2928 . . . 4 (B𝐶y(y By 𝐶))
54ralbii 2324 . . 3 (x A B𝐶x A y(y By 𝐶))
6 ralcom4 2570 . . 3 (x A y(y By 𝐶) ↔ yx A (y By 𝐶))
7 r19.23v 2419 . . . 4 (x A (y By 𝐶) ↔ (x A y By 𝐶))
87albii 1356 . . 3 (yx A (y By 𝐶) ↔ y(x A y By 𝐶))
95, 6, 83bitrri 196 . 2 (y(x A y By 𝐶) ↔ x A B𝐶)
102, 3, 93bitri 195 1 ( x A B𝐶x A B𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911  ∪ ciun 3647 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-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3649 This theorem is referenced by:  iunss2  3692  djussxp  4423  fun11iun  5088
 Copyright terms: Public domain W3C validator