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Theorem uniwun 9107
Description: Every set is contained in a weak universe. This is the analogue of grothtsk 9202 for weak universes, but it is provable in ZFC without the Tarski-Grothendieck axiom, contrary to grothtsk 9202. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
uniwun  |-  U.WUni  =  _V

Proof of Theorem uniwun
Dummy variables  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqv 3800 . 2  |-  ( U.WUni  =  _V  <->  A. x  x  e. 
U.WUni )
2 snex 4678 . . . 4  |-  { x }  e.  _V
3 wunex 9106 . . . 4  |-  ( { x }  e.  _V  ->  E. u  e. WUni  { x }  C_  u )
42, 3ax-mp 5 . . 3  |-  E. u  e. WUni  { x }  C_  u
5 eluni2 4239 . . . 4  |-  ( x  e.  U.WUni  <->  E. u  e. WUni  x  e.  u )
6 vex 3109 . . . . . 6  |-  x  e. 
_V
76snss 4140 . . . . 5  |-  ( x  e.  u  <->  { x }  C_  u )
87rexbii 2956 . . . 4  |-  ( E. u  e. WUni  x  e.  u  <->  E. u  e. WUni  { x }  C_  u )
95, 8bitri 249 . . 3  |-  ( x  e.  U.WUni  <->  E. u  e. WUni  { x }  C_  u )
104, 9mpbir 209 . 2  |-  x  e. 
U.WUni
111, 10mpgbir 1627 1  |-  U.WUni  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   E.wrex 2805   _Vcvv 3106    C_ wss 3461   {csn 4016   U.cuni 4235  WUnicwun 9067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-wun 9069
This theorem is referenced by: (None)
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