Theorem uniintsnr 3802

Description: The union and intersection of a singleton are equal. See also eusn 3592. (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 2684	. . . 4 ⊢ 𝑥 ∈ V
2	1	unisn 3747	. . 3 ⊢ ∪ {𝑥} = 𝑥
3		unieq 3740	. . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		inteq 3769	. . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥})
5	1	intsn 3801	. . . 4 ⊢ ∩ {𝑥} = 𝑥
6	4, 5	syl6eq 2186	. . 3 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥)
7	2, 3, 6	3eqtr4a 2196	. 2 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
8	7	exlimiv 1577	1 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)

Syntax hints:   → wi 4   = wceq 1331  ∃wex 1468  {csn 3522  ∪ cuni 3731  ∩ cint 3766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767

This theorem is referenced by:  uniintabim  3803