Theorem exsnrex 4067

Description: There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Assertion

Ref	Expression
exsnrex

Proof of Theorem exsnrex

Step	Hyp	Ref	Expression
1		ssnid 4058	. . . . 5
2		eleq2 2530	. . . . 5
3	1, 2	mpbiri 233	. . . 4
4	3	pm4.71ri 633	. . 3
5	4	exbii 1667	. 2
6		df-rex 2813	. 2
7	5, 6	bitr4i 252	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  {csn 4029

This theorem is referenced by:  frgrawopreg1  25050  frgrawopreg2  25051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111  df-sn 4030