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Theorem axgroth5 9848
Description: The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)
Assertion
Ref Expression
axgroth5 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem axgroth5
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 ax-groth 9847 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
2 biid 251 . . . 4 (𝑥𝑦𝑥𝑦)
3 pwss 4314 . . . . . 6 (𝒫 𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
4 pwss 4314 . . . . . . 7 (𝒫 𝑧𝑤 ↔ ∀𝑣(𝑣𝑧𝑣𝑤))
54rexbii 3189 . . . . . 6 (∃𝑤𝑦 𝒫 𝑧𝑤 ↔ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤))
63, 5anbi12i 612 . . . . 5 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)))
76ralbii 3129 . . . 4 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)))
8 df-ral 3066 . . . . 5 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)))
9 selpw 4304 . . . . . . 7 (𝑧 ∈ 𝒫 𝑦𝑧𝑦)
109imbi1i 338 . . . . . 6 ((𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
1110albii 1895 . . . . 5 (∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)) ↔ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
128, 11bitri 264 . . . 4 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
132, 7, 123anbi123i 1158 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦))))
1413exbii 1924 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦))))
151, 14mpbir 221 1 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 836  w3a 1071  wal 1629  wex 1852  wcel 2145  wral 3061  wrex 3062  wss 3723  𝒫 cpw 4297   class class class wbr 4786  cen 8106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-groth 9847
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299
This theorem is referenced by:  grothpw  9850  grothpwex  9851  axgroth6  9852  grothtsk  9859
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