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Theorem axgroth6 10866
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 10862 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
2 biid 261 . . . 4 (𝑥𝑦𝑥𝑦)
3 pweq 4619 . . . . . . . . 9 (𝑧 = 𝑣 → 𝒫 𝑧 = 𝒫 𝑣)
43sseq1d 4027 . . . . . . . 8 (𝑧 = 𝑣 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑣𝑦))
54cbvralvw 3235 . . . . . . 7 (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑣𝑦 𝒫 𝑣𝑦)
6 ssid 4018 . . . . . . . . . 10 𝒫 𝑧 ⊆ 𝒫 𝑧
7 sseq2 4022 . . . . . . . . . . 11 (𝑤 = 𝒫 𝑧 → (𝒫 𝑧𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
87rspcev 3622 . . . . . . . . . 10 ((𝒫 𝑧𝑦 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤𝑦 𝒫 𝑧𝑤)
96, 8mpan2 691 . . . . . . . . 9 (𝒫 𝑧𝑦 → ∃𝑤𝑦 𝒫 𝑧𝑤)
10 pweq 4619 . . . . . . . . . . . . 13 (𝑣 = 𝑤 → 𝒫 𝑣 = 𝒫 𝑤)
1110sseq1d 4027 . . . . . . . . . . . 12 (𝑣 = 𝑤 → (𝒫 𝑣𝑦 ↔ 𝒫 𝑤𝑦))
1211rspccv 3619 . . . . . . . . . . 11 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝑤𝑦 → 𝒫 𝑤𝑦))
13 pwss 4628 . . . . . . . . . . . 12 (𝒫 𝑤𝑦 ↔ ∀𝑣(𝑣𝑤𝑣𝑦))
14 vpwex 5383 . . . . . . . . . . . . 13 𝒫 𝑧 ∈ V
15 sseq1 4021 . . . . . . . . . . . . . 14 (𝑣 = 𝒫 𝑧 → (𝑣𝑤 ↔ 𝒫 𝑧𝑤))
16 eleq1 2827 . . . . . . . . . . . . . 14 (𝑣 = 𝒫 𝑧 → (𝑣𝑦 ↔ 𝒫 𝑧𝑦))
1715, 16imbi12d 344 . . . . . . . . . . . . 13 (𝑣 = 𝒫 𝑧 → ((𝑣𝑤𝑣𝑦) ↔ (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦)))
1814, 17spcv 3605 . . . . . . . . . . . 12 (∀𝑣(𝑣𝑤𝑣𝑦) → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
1913, 18sylbi 217 . . . . . . . . . . 11 (𝒫 𝑤𝑦 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
2012, 19syl6 35 . . . . . . . . . 10 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝑤𝑦 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦)))
2120rexlimdv 3151 . . . . . . . . 9 (∀𝑣𝑦 𝒫 𝑣𝑦 → (∃𝑤𝑦 𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
229, 21impbid2 226 . . . . . . . 8 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝒫 𝑧𝑦 ↔ ∃𝑤𝑦 𝒫 𝑧𝑤))
2322ralbidv 3176 . . . . . . 7 (∀𝑣𝑦 𝒫 𝑣𝑦 → (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
245, 23sylbi 217 . . . . . 6 (∀𝑧𝑦 𝒫 𝑧𝑦 → (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
2524pm5.32i 574 . . . . 5 ((∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ (∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
26 r19.26 3109 . . . . 5 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ↔ (∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
27 r19.26 3109 . . . . 5 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
2825, 26, 273bitr4i 303 . . . 4 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ↔ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤))
29 velpw 4610 . . . . . 6 (𝑧 ∈ 𝒫 𝑦𝑧𝑦)
30 impexp 450 . . . . . . . . 9 (((𝑧𝑦𝑧𝑦) → (¬ 𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦))))
31 ssdomg 9039 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑧𝑦𝑧𝑦))
3231elv 3483 . . . . . . . . . . 11 (𝑧𝑦𝑧𝑦)
3332pm4.71i 559 . . . . . . . . . 10 (𝑧𝑦 ↔ (𝑧𝑦𝑧𝑦))
3433imbi1i 349 . . . . . . . . 9 ((𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)) ↔ ((𝑧𝑦𝑧𝑦) → (¬ 𝑧𝑦𝑧𝑦)))
35 brsdom 9014 . . . . . . . . . . . 12 (𝑧𝑦 ↔ (𝑧𝑦 ∧ ¬ 𝑧𝑦))
3635imbi1i 349 . . . . . . . . . . 11 ((𝑧𝑦𝑧𝑦) ↔ ((𝑧𝑦 ∧ ¬ 𝑧𝑦) → 𝑧𝑦))
37 impexp 450 . . . . . . . . . . 11 (((𝑧𝑦 ∧ ¬ 𝑧𝑦) → 𝑧𝑦) ↔ (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)))
3836, 37bitri 275 . . . . . . . . . 10 ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)))
3938imbi2i 336 . . . . . . . . 9 ((𝑧𝑦 → (𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦))))
4030, 34, 393bitr4ri 304 . . . . . . . 8 ((𝑧𝑦 → (𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)))
4140pm5.74ri 272 . . . . . . 7 (𝑧𝑦 → ((𝑧𝑦𝑧𝑦) ↔ (¬ 𝑧𝑦𝑧𝑦)))
42 pm4.64 849 . . . . . . 7 ((¬ 𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦))
4341, 42bitrdi 287 . . . . . 6 (𝑧𝑦 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦)))
4429, 43sylbi 217 . . . . 5 (𝑧 ∈ 𝒫 𝑦 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦)))
4544ralbiia 3089 . . . 4 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
462, 28, 453anbi123i 1154 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)))
4746exbii 1845 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)))
481, 47mpbir 231 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 206  wa 395  wo 847  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cen 8981  cdom 8982  csdm 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-groth 10861
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-dom 8986  df-sdom 8987
This theorem is referenced by:  grothomex  10867  grothac  10868
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