Theorem el1c 4139

Description: Membership in cardinal one. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
el1c	|- (A e. 1c <-> E.x A = {x})

Distinct variable group:   x,A

Proof of Theorem el1c

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2867	. 2 |- (A e. 1c -> A e. _V)
2		snex 4111	. . . 4 |- {x} e. _V
3		eleq1 2413	. . . 4 |- (A = {x} -> (A e. _V <-> {x} e. _V))
4	2, 3	mpbiri 224	. . 3 |- (A = {x} -> A e. _V)
5	4	exlimiv 1634	. 2 |- (E.x A = {x} -> A e. _V)
6		eqeq1 2359	. . . 4 |- (y = A -> (y = {x} <-> A = {x}))
7	6	exbidv 1626	. . 3 |- (y = A -> (E.x y = {x} <-> E.x A = {x}))
8		df-1c 4136	. . 3 |- 1c = {y | E.x y = {x}}
9	7, 8	elab2g 2987	. 2 |- (A e. _V -> (A e. 1c <-> E.x A = {x}))
10	1, 5, 9	pm5.21nii 342	1 |- (A e. 1c <-> E.x A = {x})

Syntax hints:   <-> wb 176  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859  {csn 3737  1_cc1c 4134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136

This theorem is referenced by:  snel1c  4140  elpw1  4144  elpw11c  4147  0nel1c  4159  eqpw1  4162  df1c2  4168  pw111  4170  eluni1g  4172  opkelimagekg  4271  sikexlem  4295  dfimak2  4298  dfpw2  4327  eqpw1uni  4330  pw1eqadj  4332  dfeu2  4333  dfnnc2  4395  0nelsuc  4400  elsuc  4413  nnsucelrlem1  4424  nnsucelr  4428  ssfin  4470  nnadjoinlem1  4519  sfintfinlem1  4531  spfinex  4537  elimapw11c  4948  pw1fnf1o  5855  1cnc  6139  el2c  6191