Theorem unipr 4881

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) (Proof shortened by BJ, 1-Sep-2024.)

Hypotheses

Ref	Expression
unipr.1	⊢ 𝐴 ∈ V
unipr.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
unipr	⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Proof of Theorem unipr

Step	Hyp	Ref	Expression
1		unipr.1	. 2 ⊢ 𝐴 ∈ V
2		unipr.2	. 2 ⊢ 𝐵 ∈ V
3		uniprg 4880	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 2, 3	mp2an 702	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Syntax hints:   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∪ cun 3902  {cpr 4583  ∪ cuni 4864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-sn 4582  df-pr 4584  df-uni 4865

This theorem is referenced by:  uniintsn  4942  uniop  5483  unexOLD  7724  nlim2  8454  rankxplim  9834  mrcun  17637  indistps  23051  indistps2  23052  leordtval2  23252  ex-uni  30574  mnuprdlem1  44812  mnuprdlem2  44813  mnurndlem1  44821  fouriersw  46769