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Theorem df1c2 4168
 Description: Cardinal one is the unit power class of the universe. (Contributed by SF, 29-Jan-2015.)
Assertion
Ref Expression
df1c2 1c = 1V

Proof of Theorem df1c2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexv 2873 . . 3 (y V x = {y} ↔ y x = {y})
2 elpw1 4144 . . 3 (x 1V ↔ y V x = {y})
3 el1c 4139 . . 3 (x 1cy x = {y})
41, 2, 33bitr4ri 269 . 2 (x 1cx 1V)
54eqriv 2350 1 1c = 1V
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  {csn 3737  1cc1c 4134  ℘1cpw1 4135 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-rex 2620  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-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137 This theorem is referenced by:  1cvsfin  4542  tncveqnc1fin  4544  vfinspsslem1  4550  elima1c  4947  pw1fnf1o  5855  ncpw1c  6154  ce2nc1  6193  tcncv  6226  nchoicelem19  6307
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