Theorem unex 4536

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	⊢ (𝐴 ∪ 𝐵) ∈ V

Proof of Theorem unex

Step	Hyp	Ref	Expression
1		unex.1	. . 3 ⊢ 𝐴 ∈ V
2		unex.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	unipr 3905	. 2 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)
4		prexg 4299	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
5	1, 2, 4	mp2an 426	. . 3 ⊢ {𝐴, 𝐵} ∈ V
6	5	uniex 4532	. 2 ⊢ ∪ {𝐴, 𝐵} ∈ V
7	3, 6	eqeltrri 2303	1 ⊢ (𝐴 ∪ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2200  Vcvv 2800   ∪ cun 3196  {cpr 3668  ∪ cuni 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-uni 3892

This theorem is referenced by:  unexb  4537  rdg0  6548  unen  6986  findcard2  7073  findcard2s  7074  ac6sfi  7082  sbthlemi10  7159  finomni  7333  exmidfodomrlemim  7405  nn0ex  9401  xrex  10084  xnn0nnen  10692  nninfct  12605  exmidunben  13040  strleun  13180  fngsum  13464  fnpsr  14674