Theorem uniintsnr 3642

Description: The union and intersection of a singleton are equal. See also eusn 3435. (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 2554	. . . 4 ⊢ x ∈ V
2	1	unisn 3587	. . 3 ⊢ ∪ {x} = x
3		unieq 3580	. . 3 ⊢ (A = {x} → ∪ A = ∪ {x})
4		inteq 3609	. . . 4 ⊢ (A = {x} → ∩ A = ∩ {x})
5	1	intsn 3641	. . . 4 ⊢ ∩ {x} = x
6	4, 5	syl6eq 2085	. . 3 ⊢ (A = {x} → ∩ A = x)
7	2, 3, 6	3eqtr4a 2095	. 2 ⊢ (A = {x} → ∪ A = ∩ A)
8	7	exlimiv 1486	1 ⊢ (∃x A = {x} → ∪ A = ∩ A)

Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378  {csn 3367  ∪ cuni 3571  ∩ cint 3606

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607

This theorem is referenced by:  uniintabim  3643