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Theorem axgroth6 10772
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 10768 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
2 biid 261 . . . 4 (𝑥𝑦𝑥𝑦)
3 pweq 4578 . . . . . . . . 9 (𝑧 = 𝑣 → 𝒫 𝑧 = 𝒫 𝑣)
43sseq1d 3979 . . . . . . . 8 (𝑧 = 𝑣 → (𝒫 𝑧𝑦 ↔ 𝒫 𝑣𝑦))
54cbvralvw 3224 . . . . . . 7 (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑣𝑦 𝒫 𝑣𝑦)
6 ssid 3970 . . . . . . . . . 10 𝒫 𝑧 ⊆ 𝒫 𝑧
7 sseq2 3974 . . . . . . . . . . 11 (𝑤 = 𝒫 𝑧 → (𝒫 𝑧𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
87rspcev 3583 . . . . . . . . . 10 ((𝒫 𝑧𝑦 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤𝑦 𝒫 𝑧𝑤)
96, 8mpan2 690 . . . . . . . . 9 (𝒫 𝑧𝑦 → ∃𝑤𝑦 𝒫 𝑧𝑤)
10 pweq 4578 . . . . . . . . . . . . 13 (𝑣 = 𝑤 → 𝒫 𝑣 = 𝒫 𝑤)
1110sseq1d 3979 . . . . . . . . . . . 12 (𝑣 = 𝑤 → (𝒫 𝑣𝑦 ↔ 𝒫 𝑤𝑦))
1211rspccv 3580 . . . . . . . . . . 11 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝑤𝑦 → 𝒫 𝑤𝑦))
13 pwss 4587 . . . . . . . . . . . 12 (𝒫 𝑤𝑦 ↔ ∀𝑣(𝑣𝑤𝑣𝑦))
14 vpwex 5336 . . . . . . . . . . . . 13 𝒫 𝑧 ∈ V
15 sseq1 3973 . . . . . . . . . . . . . 14 (𝑣 = 𝒫 𝑧 → (𝑣𝑤 ↔ 𝒫 𝑧𝑤))
16 eleq1 2822 . . . . . . . . . . . . . 14 (𝑣 = 𝒫 𝑧 → (𝑣𝑦 ↔ 𝒫 𝑧𝑦))
1715, 16imbi12d 345 . . . . . . . . . . . . 13 (𝑣 = 𝒫 𝑧 → ((𝑣𝑤𝑣𝑦) ↔ (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦)))
1814, 17spcv 3566 . . . . . . . . . . . 12 (∀𝑣(𝑣𝑤𝑣𝑦) → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
1913, 18sylbi 216 . . . . . . . . . . 11 (𝒫 𝑤𝑦 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
2012, 19syl6 35 . . . . . . . . . 10 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝑤𝑦 → (𝒫 𝑧𝑤 → 𝒫 𝑧𝑦)))
2120rexlimdv 3147 . . . . . . . . 9 (∀𝑣𝑦 𝒫 𝑣𝑦 → (∃𝑤𝑦 𝒫 𝑧𝑤 → 𝒫 𝑧𝑦))
229, 21impbid2 225 . . . . . . . 8 (∀𝑣𝑦 𝒫 𝑣𝑦 → (𝒫 𝑧𝑦 ↔ ∃𝑤𝑦 𝒫 𝑧𝑤))
2322ralbidv 3171 . . . . . . 7 (∀𝑣𝑦 𝒫 𝑣𝑦 → (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
245, 23sylbi 216 . . . . . 6 (∀𝑧𝑦 𝒫 𝑧𝑦 → (∀𝑧𝑦 𝒫 𝑧𝑦 ↔ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
2524pm5.32i 576 . . . . 5 ((∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ (∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦𝑤𝑦 𝒫 𝑧𝑤))
26 r19.26 3111 . . . . 5 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ↔ (∀𝑧𝑦 𝒫 𝑧𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
27 r19.26 3111 . . . . 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 4569 . . . . . 6 (𝑧 ∈ 𝒫 𝑦𝑧𝑦)
30 impexp 452 . . . . . . . . 9 (((𝑧𝑦𝑧𝑦) → (¬ 𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦))))
31 ssdomg 8946 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑧𝑦𝑧𝑦))
3231elv 3453 . . . . . . . . . . 11 (𝑧𝑦𝑧𝑦)
3332pm4.71i 561 . . . . . . . . . 10 (𝑧𝑦 ↔ (𝑧𝑦𝑧𝑦))
3433imbi1i 350 . . . . . . . . 9 ((𝑧𝑦 → (¬ 𝑧𝑦𝑧𝑦)) ↔ ((𝑧𝑦𝑧𝑦) → (¬ 𝑧𝑦𝑧𝑦)))
35 brsdom 8921 . . . . . . . . . . . 12 (𝑧𝑦 ↔ (𝑧𝑦 ∧ ¬ 𝑧𝑦))
3635imbi1i 350 . . . . . . . . . . 11 ((𝑧𝑦𝑧𝑦) ↔ ((𝑧𝑦 ∧ ¬ 𝑧𝑦) → 𝑧𝑦))
37 impexp 452 . . . . . . . . . . 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 pm4.64 848 . . . . . . 7 ((¬ 𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦))
4341, 42bitrdi 287 . . . . . 6 (𝑧𝑦 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦)))
4429, 43sylbi 216 . . . . 5 (𝑧 ∈ 𝒫 𝑦 → ((𝑧𝑦𝑧𝑦) ↔ (𝑧𝑦𝑧𝑦)))
4544ralbiia 3091 . . . 4 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
462, 28, 453anbi123i 1156 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)))
4746exbii 1851 . 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 230 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 205  wa 397  wo 846  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  wral 3061  wrex 3070  Vcvv 3447  wss 3914  𝒫 cpw 4564   class class class wbr 5109  cen 8886  cdom 8887  csdm 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-groth 10767
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-dom 8891  df-sdom 8892
This theorem is referenced by:  grothomex  10773  grothac  10774
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