Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-unexg Structured version   GIF version

Theorem bj-unexg 9352
 Description: unexg 4144 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-unexg ((A 𝑉 B 𝑊) → (AB) V)

Proof of Theorem bj-unexg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3084 . . 3 (x = A → (xy) = (Ay))
2 eleq1 2097 . . 3 ((xy) = (Ay) → ((xy) V ↔ (Ay) V))
31, 2syl 14 . 2 (x = A → ((xy) V ↔ (Ay) V))
4 uneq2 3085 . . 3 (y = B → (Ay) = (AB))
5 eleq1 2097 . . 3 ((Ay) = (AB) → ((Ay) V ↔ (AB) V))
64, 5syl 14 . 2 (y = B → ((Ay) V ↔ (AB) V))
7 vex 2554 . . 3 x V
8 vex 2554 . . 3 y V
97, 8bj-unex 9350 . 2 (xy) V
103, 6, 9vtocl2g 2611 1 ((A 𝑉 B 𝑊) → (AB) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909 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-pr 3935  ax-un 4136  ax-bd0 9248  ax-bdor 9251  ax-bdex 9254  ax-bdeq 9255  ax-bdel 9256  ax-bdsb 9257  ax-bdsep 9319 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-sn 3373  df-pr 3374  df-uni 3572  df-bdc 9276 This theorem is referenced by:  bj-sucexg  9353
 Copyright terms: Public domain W3C validator