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Theorem grothtsk 9213
 Description: The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
Assertion
Ref Expression
grothtsk

Proof of Theorem grothtsk
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 9202 . . . . 5
2 vex 3116 . . . . . . . . 9
3 eltskg 9128 . . . . . . . . 9
42, 3ax-mp 5 . . . . . . . 8
54anbi2i 694 . . . . . . 7
6 3anass 977 . . . . . . 7
75, 6bitr4i 252 . . . . . 6
87exbii 1644 . . . . 5
91, 8mpbir 209 . . . 4
10 eluni 4248 . . . 4
119, 10mpbir 209 . . 3
12 vex 3116 . . 3
1311, 122th 239 . 2
1413eqriv 2463 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wo 368   wa 369   w3a 973   wceq 1379  wex 1596   wcel 1767  wral 2814  wrex 2815  cvv 3113   wss 3476  cpw 4010  cuni 4245   class class class wbr 4447   cen 7513  ctsk 9126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-groth 9201 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-tsk 9127 This theorem is referenced by:  inaprc  9214  tskmval  9217  tskmcl  9219
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