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

Theorem unexb 4143
Description: Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
Assertion
Ref Expression
unexb ((A V B V) ↔ (AB) V)

Proof of Theorem unexb
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3084 . . . 4 (x = A → (xy) = (Ay))
21eleq1d 2103 . . 3 (x = A → ((xy) V ↔ (Ay) V))
3 uneq2 3085 . . . 4 (y = B → (Ay) = (AB))
43eleq1d 2103 . . 3 (y = B → ((Ay) V ↔ (AB) V))
5 vex 2554 . . . 4 x V
6 vex 2554 . . . 4 y V
75, 6unex 4142 . . 3 (xy) V
82, 4, 7vtocl2g 2611 . 2 ((A V B V) → (AB) V)
9 ssun1 3100 . . . 4 A ⊆ (AB)
10 ssexg 3887 . . . 4 ((A ⊆ (AB) (AB) V) → A V)
119, 10mpan 400 . . 3 ((AB) V → A V)
12 ssun2 3101 . . . 4 B ⊆ (AB)
13 ssexg 3887 . . . 4 ((B ⊆ (AB) (AB) V) → B V)
1412, 13mpan 400 . . 3 ((AB) V → B V)
1511, 14jca 290 . 2 ((AB) V → (A V B V))
168, 15impbii 117 1 ((A V B V) ↔ (AB) V)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935  ax-un 4136
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-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by:  unexg  4144  sucexb  4189  frecabex  5923
  Copyright terms: Public domain W3C validator