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Theorem unex 4202
 Description: The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
Hypotheses
Ref Expression
unex.1 𝐴 ∈ V
unex.2 𝐵 ∈ V
Assertion
Ref Expression
unex (𝐴𝐵) ∈ V

Proof of Theorem unex
StepHypRef Expression
1 unex.1 . . 3 𝐴 ∈ V
2 unex.2 . . 3 𝐵 ∈ V
31, 2unipr 3623 . 2 {𝐴, 𝐵} = (𝐴𝐵)
4 prexg 3974 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
51, 2, 4mp2an 417 . . 3 {𝐴, 𝐵} ∈ V
65uniex 4200 . 2 {𝐴, 𝐵} ∈ V
73, 6eqeltrri 2153 1 (𝐴𝐵) ∈ V
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1434  Vcvv 2602   ∪ cun 2972  {cpr 3407  ∪ 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-sep 3904  ax-pr 3972  ax-un 4196 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-un 2978  df-sn 3412  df-pr 3413  df-uni 3610 This theorem is referenced by:  unexb  4203  rdg0  6036  unen  6361  findcard2  6423  findcard2s  6424  ac6sfi  6431  nn0ex  8361  xrex  8986
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