MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  griedg0prc Structured version   Visualization version   GIF version

Theorem griedg0prc 28254
Description: The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.)
Hypothesis
Ref Expression
griedg0prc.u π‘ˆ = {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…}
Assertion
Ref Expression
griedg0prc π‘ˆ βˆ‰ V
Distinct variable group:   𝑣,𝑒
Allowed substitution hints:   π‘ˆ(𝑣,𝑒)

Proof of Theorem griedg0prc
StepHypRef Expression
1 0ex 5265 . . . 4 βˆ… ∈ V
2 feq1 6650 . . . 4 (𝑒 = βˆ… β†’ (𝑒:βˆ…βŸΆβˆ… ↔ βˆ…:βˆ…βŸΆβˆ…))
3 f0 6724 . . . 4 βˆ…:βˆ…βŸΆβˆ…
41, 2, 3ceqsexv2d 3496 . . 3 βˆƒπ‘’ 𝑒:βˆ…βŸΆβˆ…
5 opabn1stprc 7991 . . 3 (βˆƒπ‘’ 𝑒:βˆ…βŸΆβˆ… β†’ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V)
64, 5ax-mp 5 . 2 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V
7 griedg0prc.u . . 3 π‘ˆ = {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…}
8 neleq1 3051 . . 3 (π‘ˆ = {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} β†’ (π‘ˆ βˆ‰ V ↔ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V))
97, 8ax-mp 5 . 2 (π‘ˆ βˆ‰ V ↔ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V)
106, 9mpbir 230 1 π‘ˆ βˆ‰ V
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1542  βˆƒwex 1782   βˆ‰ wnel 3046  Vcvv 3444  βˆ…c0 4283  {copab 5168  βŸΆwf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  usgrprc  28256  rgrusgrprc  28579
  Copyright terms: Public domain W3C validator