Theorem axgroth6 10584

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(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axgroth6

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axgroth5 10580	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
2		biid 260	. . . 4 ⊢ (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)
3		pweq 4549	. . . . . . . . 9 ⊢ (𝑧 = 𝑣 → 𝒫 𝑧 = 𝒫 𝑣)
4	3	sseq1d 3952	. . . . . . . 8 ⊢ (𝑧 = 𝑣 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑣 ⊆ 𝑦))
5	4	cbvralvw 3383	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ↔ ∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦)
6		ssid 3943	. . . . . . . . . 10 ⊢ 𝒫 𝑧 ⊆ 𝒫 𝑧
7		sseq2 3947	. . . . . . . . . . 11 ⊢ (𝑤 = 𝒫 𝑧 → (𝒫 𝑧 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
8	7	rspcev 3561	. . . . . . . . . 10 ⊢ ((𝒫 𝑧 ∈ 𝑦 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)
9	6, 8	mpan2 688	. . . . . . . . 9 ⊢ (𝒫 𝑧 ∈ 𝑦 → ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)
10		pweq 4549	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → 𝒫 𝑣 = 𝒫 𝑤)
11	10	sseq1d 3952	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (𝒫 𝑣 ⊆ 𝑦 ↔ 𝒫 𝑤 ⊆ 𝑦))
12	11	rspccv 3558	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (𝑤 ∈ 𝑦 → 𝒫 𝑤 ⊆ 𝑦))
13		pwss 4558	. . . . . . . . . . . 12 ⊢ (𝒫 𝑤 ⊆ 𝑦 ↔ ∀𝑣(𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑦))
14		vpwex 5300	. . . . . . . . . . . . 13 ⊢ 𝒫 𝑧 ∈ V
15		sseq1 3946	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝒫 𝑧 → (𝑣 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝑤))
16		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝒫 𝑧 → (𝑣 ∈ 𝑦 ↔ 𝒫 𝑧 ∈ 𝑦))
17	15, 16	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝒫 𝑧 → ((𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑦) ↔ (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦)))
18	14, 17	spcv 3544	. . . . . . . . . . . 12 ⊢ (∀𝑣(𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑦) → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦))
19	13, 18	sylbi 216	. . . . . . . . . . 11 ⊢ (𝒫 𝑤 ⊆ 𝑦 → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦))
20	12, 19	syl6 35	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (𝑤 ∈ 𝑦 → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦)))
21	20	rexlimdv 3212	. . . . . . . . 9 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦))
22	9, 21	impbid2 225	. . . . . . . 8 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (𝒫 𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
23	22	ralbidv 3112	. . . . . . 7 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 ↔ ∀𝑧 ∈ 𝑦 ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
24	5, 23	sylbi 216	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 ↔ ∀𝑧 ∈ 𝑦 ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
25	24	pm5.32i 575	. . . . 5 ⊢ ((∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦) ↔ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
26		r19.26 3095	. . . . 5 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ↔ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
27		r19.26 3095	. . . . 5 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
28	25, 26, 27	3bitr4i 303	. . . 4 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
29		velpw 4538	. . . . . 6 ⊢ (𝑧 ∈ 𝒫 𝑦 ↔ 𝑧 ⊆ 𝑦)
30		impexp 451	. . . . . . . . 9 ⊢ (((𝑧 ⊆ 𝑦 ∧ 𝑧 ≼ 𝑦) → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)) ↔ (𝑧 ⊆ 𝑦 → (𝑧 ≼ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦))))
31		ssdomg 8786	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑧 ⊆ 𝑦 → 𝑧 ≼ 𝑦))
32	31	elv 3438	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝑦 → 𝑧 ≼ 𝑦)
33	32	pm4.71i 560	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ⊆ 𝑦 ∧ 𝑧 ≼ 𝑦))
34	33	imbi1i 350	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)) ↔ ((𝑧 ⊆ 𝑦 ∧ 𝑧 ≼ 𝑦) → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
35		brsdom 8763	. . . . . . . . . . . 12 ⊢ (𝑧 ≺ 𝑦 ↔ (𝑧 ≼ 𝑦 ∧ ¬ 𝑧 ≈ 𝑦))
36	35	imbi1i 350	. . . . . . . . . . 11 ⊢ ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ ((𝑧 ≼ 𝑦 ∧ ¬ 𝑧 ≈ 𝑦) → 𝑧 ∈ 𝑦))
37		impexp 451	. . . . . . . . . . 11 ⊢ (((𝑧 ≼ 𝑦 ∧ ¬ 𝑧 ≈ 𝑦) → 𝑧 ∈ 𝑦) ↔ (𝑧 ≼ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
38	36, 37	bitri 274	. . . . . . . . . 10 ⊢ ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≼ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
39	38	imbi2i 336	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝑦 → (𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) ↔ (𝑧 ⊆ 𝑦 → (𝑧 ≼ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦))))
40	30, 34, 39	3bitr4ri 304	. . . . . . . 8 ⊢ ((𝑧 ⊆ 𝑦 → (𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) ↔ (𝑧 ⊆ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
41	40	pm5.74ri 271	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑦 → ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
42		pm4.64 846	. . . . . . 7 ⊢ ((¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
43	41, 42	bitrdi 287	. . . . . 6 ⊢ (𝑧 ⊆ 𝑦 → ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
44	29, 43	sylbi 216	. . . . 5 ⊢ (𝑧 ∈ 𝒫 𝑦 → ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
45	44	ralbiia 3091	. . . 4 ⊢ (∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
46	2, 28, 45	3anbi123i 1154	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
47	46	exbii 1850	. 2 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
48	1, 47	mpbir 230	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074   ≈ cen 8730   ≼ cdom 8731   ≺ csdm 8732

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-groth 10579

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-dom 8735  df-sdom 8736

This theorem is referenced by:  grothomex  10585  grothac  10586