Theorem df1c2 4169

Description: Cardinal one is the unit power class of the universe. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
df1c2	|- 1c = ~P1 _V

Proof of Theorem df1c2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexv 2874	. . 3 |- (E.y e. _V x = {y} <-> E.y x = {y})
2		elpw1 4145	. . 3 |- (x e. ~P1 _V <-> E.y e. _V x = {y})
3		el1c 4140	. . 3 |- (x e. 1c <-> E.y x = {y})
4	1, 2, 3	3bitr4ri 269	. 2 |- (x e. 1c <-> x e. ~P1 _V)
5	4	eqriv 2350	1 |- 1c = ~P1 _V

Syntax hints:  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860  {csn 3738  1_cc1c 4135  ~P₁ cpw1 4136

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-1c 4137  df-pw1 4138

This theorem is referenced by:  1cvsfin  4543  tncveqnc1fin  4545  vfinspsslem1  4551  elima1c  4948  pw1fnf1o  5856  ncpw1c  6155  ce2nc1  6194  tcncv  6227  nchoicelem19  6308