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Theorem griedg0prc 29121
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 5302 . . . 4 βˆ… ∈ V
2 feq1 6698 . . . 4 (𝑒 = βˆ… β†’ (𝑒:βˆ…βŸΆβˆ… ↔ βˆ…:βˆ…βŸΆβˆ…))
3 f0 6773 . . . 4 βˆ…:βˆ…βŸΆβˆ…
41, 2, 3ceqsexv2d 3518 . . 3 βˆƒπ‘’ 𝑒:βˆ…βŸΆβˆ…
5 opabn1stprc 8060 . . 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 3042 . . 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 1533  βˆƒwex 1773   βˆ‰ wnel 3036  Vcvv 3463  βˆ…c0 4318  {copab 5205  βŸΆwf 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  usgrprc  29123  rgrusgrprc  29447
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