Theorem epprc 5828

Description: The membership relationship is a proper class. This theorem together with vvex 4110 demonstrates the basic idea behind New Foundations: since x e. 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 e. _V

Proof of Theorem epprc

Step	Hyp	Ref	Expression
1		ru 3046	. . 3 |- {x | x e/ x} e/ _V
2		df-nel 2520	. . 3 |- ({x | x e/ x} e/ _V <-> -. {x | x e/ x} e. _V)
3	1, 2	mpbi 199	. 2 |- -. {x | x e/ x} e. _V
4		elfix 5788	. . . . . . 7 |- (x e. Fix _E <-> x _E x)
5		epel 4767	. . . . . . 7 |- (x _E x <-> x e. x)
6	4, 5	bitri 240	. . . . . 6 |- (x e. Fix _E <-> x e. x)
7	6	notbii 287	. . . . 5 |- (-. x e. Fix _E <-> -. x e. x)
8		vex 2863	. . . . . 6 |- x e. _V
9	8	elcompl 3226	. . . . 5 |- (x e. ∼ Fix _E <-> -. x e. Fix _E )
10		df-nel 2520	. . . . 5 |- (x e/ x <-> -. x e. x)
11	7, 9, 10	3bitr4i 268	. . . 4 |- (x e. ∼ Fix _E <-> x e/ x)
12	11	abbi2i 2465	. . 3 |- ∼ Fix _E = {x | x e/ x}
13		fixexg 5789	. . . 4 |- ( _E e. _V -> Fix _E e. _V)
14		complexg 4100	. . . 4 |- (Fix _E e. _V -> ∼ Fix _E e. _V)
15	13, 14	syl 15	. . 3 |- ( _E e. _V -> ∼ Fix _E e. _V)
16	12, 15	syl5eqelr 2438	. 2 |- ( _E e. _V -> {x | x e/ x} e. _V)
17	3, 16	mto 167	1 |- -. _E e. _V

Syntax hints:  -. wn 3   e. wcel 1710  {cab 2339   e/ wnel 2518  _Vcvv 2860   ∼ ccompl 3206   class class class wbr 4640   _E cep 4763  Fixcfix 5740

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-nel 2520  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-swap 4725  df-sset 4726  df-ima 4728  df-eprel 4765  df-id 4768  df-cnv 4786  df-rn 4787  df-fix 5741

This theorem is referenced by: (None)