Theorem unipr 4929

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 4928	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 2, 3	mp2an 692	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961  {cpr 4633  ∪ cuni 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634  df-uni 4913

This theorem is referenced by:  uniintsn  4990  uniop  5525  unexOLD  7764  nlim2  8527  rankxplim  9917  mrcun  17667  indistps  23034  indistps2  23035  leordtval2  23236  ex-uni  30455  mnuprdlem1  44268  mnuprdlem2  44269  mnurndlem1  44277  fouriersw  46187