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Theorem epprc 5827
 Description: The membership relationship is a proper class. This theorem together with vvex 4109 demonstrates the basic idea behind New Foundations: since x ∈ y is not a stratified relationship, then it does not have a realization as a set of ordered pairs, but since x = x is stratified, then it does have a realization as a set. (Contributed by SF, 20-Feb-2015.)
Assertion
Ref Expression
epprc ¬ E V

Proof of Theorem epprc
StepHypRef Expression
1 ru 3045 . . 3 {x x x} V
2 df-nel 2519 . . 3 ({x x x} V ↔ ¬ {x x x} V)
31, 2mpbi 199 . 2 ¬ {x x x} V
4 elfix 5787 . . . . . . 7 (x Fix E ↔ x E x)
5 epel 4766 . . . . . . 7 (x E xx x)
64, 5bitri 240 . . . . . 6 (x Fix E ↔ x x)
76notbii 287 . . . . 5 x Fix E ↔ ¬ x x)
8 vex 2862 . . . . . 6 x V
98elcompl 3225 . . . . 5 (x Fix E ↔ ¬ x Fix E )
10 df-nel 2519 . . . . 5 (x x ↔ ¬ x x)
117, 9, 103bitr4i 268 . . . 4 (x Fix E ↔ x x)
1211abbi2i 2464 . . 3 Fix E = {x x x}
13 fixexg 5788 . . . 4 ( E V → Fix E V)
14 complexg 4099 . . . 4 ( Fix E V → ∼ Fix E V)
1513, 14syl 15 . . 3 ( E V → ∼ Fix E V)
1612, 15syl5eqelr 2438 . 2 ( E V → {x x x} V)
173, 16mto 167 1 ¬ E V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1710  {cab 2339   ∉ wnel 2517  Vcvv 2859   ∼ ccompl 3205   class class class wbr 4639   E cep 4762   Fix cfix 5739 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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-nel 2519  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-swap 4724  df-sset 4725  df-ima 4727  df-eprel 4764  df-id 4767  df-cnv 4785  df-rn 4786  df-fix 5740 This theorem is referenced by: (None)
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