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

Theorem bj-uniex 10866
Description: uniex 4200 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-uniex.1 𝐴 ∈ V
Assertion
Ref Expression
bj-uniex 𝐴 ∈ V

Proof of Theorem bj-uniex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-uniex.1 . 2 𝐴 ∈ V
2 unieq 3618 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
32eleq1d 2148 . 2 (𝑥 = 𝐴 → ( 𝑥 ∈ V ↔ 𝐴 ∈ V))
4 bj-uniex2 10865 . . 3 𝑦 𝑦 = 𝑥
54issetri 2609 . 2 𝑥 ∈ V
61, 3, 5vtocl 2654 1 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  Vcvv 2602   cuni 3609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-un 4196  ax-bd0 10762  ax-bdex 10768  ax-bdel 10770  ax-bdsb 10771  ax-bdsep 10833
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-uni 3610  df-bdc 10790
This theorem is referenced by:  bj-uniexg  10867  bj-unex  10868
  Copyright terms: Public domain W3C validator