Theorem uniintsnr 3882

Description: The union and intersection of a singleton are equal. See also eusn 3668. (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 2742	. . . 4 ⊢ 𝑥 ∈ V
2	1	unisn 3827	. . 3 ⊢ ∪ {𝑥} = 𝑥
3		unieq 3820	. . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		inteq 3849	. . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥})
5	1	intsn 3881	. . . 4 ⊢ ∩ {𝑥} = 𝑥
6	4, 5	eqtrdi 2226	. . 3 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥)
7	2, 3, 6	3eqtr4a 2236	. 2 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
8	7	exlimiv 1598	1 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)

Syntax hints:   → wi 4   = wceq 1353  ∃wex 1492  {csn 3594  ∪ cuni 3811  ∩ cint 3846

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847

This theorem is referenced by:  uniintabim  3883