Theorem unipr 4885

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 4884	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 2, 3	mp2an 704	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Syntax hints:   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∪ cun 3905  {cpr 4587  ∪ cuni 4868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-sn 4586  df-pr 4588  df-uni 4869

This theorem is referenced by:  uniintsn  4946  uniop  5489  unexOLD  7732  nlim2  8463  rankxplim  9839  mrcun  17668  indistps  23129  indistps2  23130  leordtval2  23330  ex-uni  30686  mnuprdlem1  44846  mnuprdlem2  44847  mnurndlem1  44855  fouriersw  46803