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Theorem axgroth6 9851
Description: The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set 𝑥, there exists a set 𝑦 containing 𝑥, the subsets of the members of 𝑦, the power sets of the members of 𝑦, and the subsets of 𝑦 of cardinality less than that of 𝑦. (Contributed by NM, 21-Jun-2009.)
Assertion
Ref Expression
axgroth6 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axgroth6
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 9847 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
2 biid 251 . . . 4 (𝑥𝑦𝑥𝑦)
3 pweq 4298 . . . . . . . . 9 (𝑧 = 𝑣 → 𝒫 𝑧 = 𝒫 𝑣)
43sseq1d 3779 . . . . . . . 8 (𝑧 = 𝑣 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑣𝑦))
54cbvralv 3319 . . . . . . 7 (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑣𝑦 𝒫 𝑣𝑦)
6 ssid 3771 . . . . . . . . . 10 𝒫 𝑧 ⊆ 𝒫 𝑧
7 sseq2 3774 . . . . . . . . . . 11 (𝑤 = 𝒫 𝑧 → (𝒫 𝑧𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
87rspcev 3458 . . . . . . . . . 10 ((𝒫 𝑧𝑦 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤𝑦 𝒫 𝑧𝑤)
96, 8mpan2 663 . . . . . . . . 9 (𝒫 𝑧𝑦 → ∃𝑤𝑦 𝒫 𝑧𝑤)
10 pweq 4298 . . . . . . . . . . . . 13 (𝑣 = 𝑤 → 𝒫 𝑣 = 𝒫 𝑤)
1110sseq1d 3779 . . . . . . . . . . . 12 (𝑣 = 𝑤 → (𝒫 𝑣𝑦 ↔ 𝒫 𝑤𝑦))
1211rspccv 3455 . . . . . . . . . . 11 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝑤𝑦 → 𝒫 𝑤𝑦))
13 pwss 4312 . . . . . . . . . . . 12 (𝒫 𝑤𝑦 ↔ ∀𝑣(𝑣𝑤𝑣𝑦))
14 vpwex 4977 . . . . . . . . . . . . 13 𝒫 𝑧 ∈ V
15 sseq1 3773 . . . . . . . . . . . . . 14 (𝑣 = 𝒫 𝑧 → (𝑣𝑤 ↔ 𝒫 𝑧𝑤))
16 eleq1 2837 . . . . . . . . . . . . . 14 (𝑣 = 𝒫 𝑧 → (𝑣𝑦 ↔ 𝒫 𝑧𝑦))
1715, 16imbi12d 333 . . . . . . . . . . . . 13 (𝑣 = 𝒫 𝑧 → ((𝑣𝑤𝑣𝑦) ↔ (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦)))
1814, 17spcv 3448 . . . . . . . . . . . 12 (∀𝑣(𝑣𝑤𝑣𝑦) → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
1913, 18sylbi 207 . . . . . . . . . . 11 (𝒫 𝑤𝑦 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
2012, 19syl6 35 . . . . . . . . . 10 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝑤𝑦 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦)))
2120rexlimdv 3177 . . . . . . . . 9 (∀𝑣𝑦 𝒫 𝑣𝑦 → (∃𝑤𝑦 𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
229, 21impbid2 216 . . . . . . . 8 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝒫 𝑧𝑦 ↔ ∃𝑤𝑦 𝒫 𝑧𝑤))
2322ralbidv 3134 . . . . . . 7 (∀𝑣𝑦 𝒫 𝑣𝑦 → (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
245, 23sylbi 207 . . . . . 6 (∀𝑧𝑦 𝒫 𝑧𝑦 → (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
2524pm5.32i 556 . . . . 5 ((∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ (∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
26 r19.26 3211 . . . . 5 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ↔ (∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
27 r19.26 3211 . . . . 5 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
2825, 26, 273bitr4i 292 . . . 4 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ↔ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤))
29 selpw 4302 . . . . . 6 (𝑧 ∈ 𝒫 𝑦𝑧𝑦)
30 impexp 437 . . . . . . . . 9 (((𝑧𝑦𝑧𝑦) → (¬ 𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦))))
31 vex 3352 . . . . . . . . . . . 12 𝑦 ∈ V
32 ssdomg 8154 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑧𝑦𝑧𝑦))
3331, 32ax-mp 5 . . . . . . . . . . 11 (𝑧𝑦𝑧𝑦)
3433pm4.71i 541 . . . . . . . . . 10 (𝑧𝑦 ↔ (𝑧𝑦𝑧𝑦))
3534imbi1i 338 . . . . . . . . 9 ((𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)) ↔ ((𝑧𝑦𝑧𝑦) → (¬ 𝑧𝑦𝑧𝑦)))
36 brsdom 8131 . . . . . . . . . . . 12 (𝑧𝑦 ↔ (𝑧𝑦 ∧ ¬ 𝑧𝑦))
3736imbi1i 338 . . . . . . . . . . 11 ((𝑧𝑦𝑧𝑦) ↔ ((𝑧𝑦 ∧ ¬ 𝑧𝑦) → 𝑧𝑦))
38 impexp 437 . . . . . . . . . . 11 (((𝑧𝑦 ∧ ¬ 𝑧𝑦) → 𝑧𝑦) ↔ (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)))
3937, 38bitri 264 . . . . . . . . . 10 ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)))
4039imbi2i 325 . . . . . . . . 9 ((𝑧𝑦 → (𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦))))
4130, 35, 403bitr4ri 293 . . . . . . . 8 ((𝑧𝑦 → (𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)))
4241pm5.74ri 261 . . . . . . 7 (𝑧𝑦 → ((𝑧𝑦𝑧𝑦) ↔ (¬ 𝑧𝑦𝑧𝑦)))
43 pm4.64 828 . . . . . . 7 ((¬ 𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦))
4442, 43syl6bb 276 . . . . . 6 (𝑧𝑦 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦)))
4529, 44sylbi 207 . . . . 5 (𝑧 ∈ 𝒫 𝑦 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦)))
4645ralbiia 3127 . . . 4 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
472, 28, 463anbi123i 1157 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)))
4847exbii 1923 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)))
491, 48mpbir 221 1 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  w3a 1070  wal 1628   = wceq 1630  wex 1851  wcel 2144  wral 3060  wrex 3061  Vcvv 3349  wss 3721  𝒫 cpw 4295   class class class wbr 4784  cen 8105  cdom 8106  csdm 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-groth 9846
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-dom 8110  df-sdom 8111
This theorem is referenced by:  grothomex  9852  grothac  9853
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