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Theorem axgroth5 9981
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 9980 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
2 biid 253 . . . 4 (𝑥𝑦𝑥𝑦)
3 pwss 4396 . . . . . 6 (𝒫 𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
4 pwss 4396 . . . . . . 7 (𝒫 𝑧𝑤 ↔ ∀𝑣(𝑣𝑧𝑣𝑤))
54rexbii 3224 . . . . . 6 (∃𝑤𝑦 𝒫 𝑧𝑤 ↔ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤))
63, 5anbi12i 620 . . . . 5 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)))
76ralbii 3162 . . . 4 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)))
8 df-ral 3095 . . . . 5 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)))
9 selpw 4386 . . . . . . 7 (𝑧 ∈ 𝒫 𝑦𝑧𝑦)
109imbi1i 341 . . . . . 6 ((𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
1110albii 1863 . . . . 5 (∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)) ↔ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
128, 11bitri 267 . . . 4 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
132, 7, 123anbi123i 1155 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦))))
1413exbii 1892 . 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 223 1 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wo 836  w3a 1071  wal 1599  wex 1823  wcel 2107  wral 3090  wrex 3091  wss 3792  𝒫 cpw 4379   class class class wbr 4886  cen 8238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-groth 9980
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-in 3799  df-ss 3806  df-pw 4381
This theorem is referenced by:  grothpw  9983  grothpwex  9984  axgroth6  9985  grothtsk  9992
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