Theorem unipr 4887

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 4886	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 2, 3	mp2an 705	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∪ cun 3900  {cpr 4589  ∪ cuni 4870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-sn 4588  df-pr 4590  df-uni 4871

This theorem is used by:  uniintsn  4948  uniop  5496  nlim2  8481  rankxplim  9865  mrcun  17716  indistps  23242  indistps2  23243  leordtval2  23443  ex-uni  30914  mnuprdlem1  45104  mnuprdlem2  45105  mnurndlem1  45113  fouriersw  47067