Theorem uniintsnr 3958

Description: The union and intersection of a singleton are equal. See also eusn 3740. (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 2802	. . . 4 ⊢ 𝑥 ∈ V
2	1	unisn 3903	. . 3 ⊢ ∪ {𝑥} = 𝑥
3		unieq 3896	. . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		inteq 3925	. . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥})
5	1	intsn 3957	. . . 4 ⊢ ∩ {𝑥} = 𝑥
6	4, 5	eqtrdi 2278	. . 3 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥)
7	2, 3, 6	3eqtr4a 2288	. 2 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
8	7	exlimiv 1644	1 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)

Syntax hints:   → wi 4   = wceq 1395  ∃wex 1538  {csn 3666  ∪ cuni 3887  ∩ cint 3922

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923

This theorem is referenced by:  uniintabim  3959