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Theorem grothtsk 9601
Description: The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
Assertion
Ref Expression
grothtsk Tarski = V

Proof of Theorem grothtsk
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 9590 . . . . 5 𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))
2 vex 3189 . . . . . . . . 9 𝑥 ∈ V
3 eltskg 9516 . . . . . . . . 9 (𝑥 ∈ V → (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
42, 3ax-mp 5 . . . . . . . 8 (𝑥 ∈ Tarski ↔ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
54anbi2i 729 . . . . . . 7 ((𝑤𝑥𝑥 ∈ Tarski) ↔ (𝑤𝑥 ∧ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
6 3anass 1040 . . . . . . 7 ((𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)) ↔ (𝑤𝑥 ∧ (∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥))))
75, 6bitr4i 267 . . . . . 6 ((𝑤𝑥𝑥 ∈ Tarski) ↔ (𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
87exbii 1771 . . . . 5 (∃𝑥(𝑤𝑥𝑥 ∈ Tarski) ↔ ∃𝑥(𝑤𝑥 ∧ ∀𝑦𝑥 (𝒫 𝑦𝑥 ∧ ∃𝑧𝑥 𝒫 𝑦𝑧) ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦𝑥𝑦𝑥)))
91, 8mpbir 221 . . . 4 𝑥(𝑤𝑥𝑥 ∈ Tarski)
10 eluni 4405 . . . 4 (𝑤 Tarski ↔ ∃𝑥(𝑤𝑥𝑥 ∈ Tarski))
119, 10mpbir 221 . . 3 𝑤 Tarski
12 vex 3189 . . 3 𝑤 ∈ V
1311, 122th 254 . 2 (𝑤 Tarski ↔ 𝑤 ∈ V)
1413eqriv 2618 1 Tarski = V
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  cen 7896  Tarskictsk 9514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-groth 9589
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-tsk 9515
This theorem is referenced by:  inaprc  9602  tskmval  9605  tskmcl  9607
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