Theorem unipr 4905

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 4904	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 2, 3	mp2an 692	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∪ cun 3929  {cpr 4608  ∪ cuni 4888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-sn 4607  df-pr 4609  df-uni 4889

This theorem is referenced by:  uniintsn  4966  uniop  5495  unexOLD  7744  nlim2  8507  rankxplim  9898  mrcun  17639  indistps  22954  indistps2  22955  leordtval2  23155  ex-uni  30412  mnuprdlem1  44263  mnuprdlem2  44264  mnurndlem1  44272  fouriersw  46227