Theorem unex 4585

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 3947	. 2 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)
4		prexg 4347	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
5	1, 2, 4	mp2an 430	. . 3 ⊢ {𝐴, 𝐵} ∈ V
6	5	uniex 4581	. 2 ⊢ ∪ {𝐴, 𝐵} ∈ V
7	3, 6	eqeltrri 2312	1 ⊢ (𝐴 ∪ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218  {cpr 3709  ∪ cuni 3933

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-uni 3934

This theorem is referenced by:  unexb  4586  rdg0  6652  unen  7099  findcard2  7187  findcard2s  7188  ac6sfi  7196  sbthlemi10  7277  finomni  7474  exmidfodomrlemim  7547  nn0ex  9552  xrex  10241  xnn0nnen  10857  hashfibclem  11265  nninfct  12801  exmidunben  13300  strleun  13441  fngzsum  13691  fnpsr  15034