Theorem uniintsnr 3964

Description: The union and intersection of a singleton are equal. See also eusn 3745. (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 2805	. . . 4 ⊢ 𝑥 ∈ V
2	1	unisn 3909	. . 3 ⊢ ∪ {𝑥} = 𝑥
3		unieq 3902	. . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		inteq 3931	. . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥})
5	1	intsn 3963	. . . 4 ⊢ ∩ {𝑥} = 𝑥
6	4, 5	eqtrdi 2280	. . 3 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥)
7	2, 3, 6	3eqtr4a 2290	. 2 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
8	7	exlimiv 1646	1 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)

Syntax hints:   → wi 4   = wceq 1397  ∃wex 1540  {csn 3669  ∪ cuni 3893  ∩ cint 3928

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929

This theorem is referenced by:  uniintabim  3965