Theorem uniintsnr 4001

Description: The union and intersection of a singleton are equal. See also eusn 3781. (Contributed by Jim Kingdon, 14-Aug-2018.)

Assertion

Ref	Expression
uniintsnr	⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem uniintsnr

Step	Hyp	Ref	Expression
1		vex 2824	. . . 4 ⊢ 𝑥 ∈ V
2	1	unisn 3946	. . 3 ⊢ ∪ {𝑥} = 𝑥
3		unieq 3939	. . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		inteq 3968	. . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥})
5	1	intsn 4000	. . . 4 ⊢ ∩ {𝑥} = 𝑥
6	4, 5	eqtrdi 2287	. . 3 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥)
7	2, 3, 6	3eqtr4a 2297	. 2 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
8	7	exlimiv 1651	1 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)

Syntax hints:   → wi 4   = wceq 1402  ∃wex 1545  {csn 3705  ∪ cuni 3930  ∩ cint 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966

This theorem is referenced by:  uniintabim  4002