Theorem axgroth6 10749

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 10745	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
2		biid 262	. . . 4 ⊢ (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)
3		pweq 4550	. . . . . . . . 9 ⊢ (𝑧 = 𝑣 → 𝒫 𝑧 = 𝒫 𝑣)
4	3	sseq1d 3953	. . . . . . . 8 ⊢ (𝑧 = 𝑣 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑣 ⊆ 𝑦))
5	4	cbvralvw 3218	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ↔ ∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦)
6		ssid 3944	. . . . . . . . . 10 ⊢ 𝒫 𝑧 ⊆ 𝒫 𝑧
7		sseq2 3948	. . . . . . . . . . 11 ⊢ (𝑤 = 𝒫 𝑧 → (𝒫 𝑧 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
8	7	rspcev 3567	. . . . . . . . . 10 ⊢ ((𝒫 𝑧 ∈ 𝑦 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)
9	6, 8	mpan2 697	. . . . . . . . 9 ⊢ (𝒫 𝑧 ∈ 𝑦 → ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)
10		pweq 4550	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → 𝒫 𝑣 = 𝒫 𝑤)
11	10	sseq1d 3953	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (𝒫 𝑣 ⊆ 𝑦 ↔ 𝒫 𝑤 ⊆ 𝑦))
12	11	rspccv 3564	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (𝑤 ∈ 𝑦 → 𝒫 𝑤 ⊆ 𝑦))
13		pwss 4559	. . . . . . . . . . . 12 ⊢ (𝒫 𝑤 ⊆ 𝑦 ↔ ∀𝑣(𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑦))
14		vpwex 5313	. . . . . . . . . . . . 13 ⊢ 𝒫 𝑧 ∈ V
15		sseq1 3947	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝒫 𝑧 → (𝑣 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝑤))
16		eleq1 2828	. . . . . . . . . . . . . 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 3550	. . . . . . . . . . . 12 ⊢ (∀𝑣(𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑦) → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦))
19	13, 18	sylbi 218	. . . . . . . . . . 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 3139	. . . . . . . . 9 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦))
22	9, 21	impbid2 227	. . . . . . . 8 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (𝒫 𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
23	22	ralbidv 3163	. . . . . . 7 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 ↔ ∀𝑧 ∈ 𝑦 ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
24	5, 23	sylbi 218	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 ↔ ∀𝑧 ∈ 𝑦 ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
25	24	pm5.32i 579	. . . . 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 3100	. . . . 5 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ↔ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
27		r19.26 3100	. . . . 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 304	. . . 4 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
29		velpw 4541	. . . . . 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 8944	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑧 ⊆ 𝑦 → 𝑧 ≼ 𝑦))
32	31	elv 3437	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝑦 → 𝑧 ≼ 𝑦)
33	32	pm4.71i 564	. . . . . . . . . 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 8918	. . . . . . . . . . . 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 276	. . . . . . . . . 10 ⊢ ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≼ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
39	38	imbi2i 337	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝑦 → (𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) ↔ (𝑧 ⊆ 𝑦 → (𝑧 ≼ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦))))
40	30, 34, 39	3bitr4ri 305	. . . . . . . 8 ⊢ ((𝑧 ⊆ 𝑦 → (𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) ↔ (𝑧 ⊆ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
41	40	pm5.74ri 273	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑦 → ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
42		pm4.64 855	. . . . . . 7 ⊢ ((¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
43	41, 42	bitrdi 288	. . . . . 6 ⊢ (𝑧 ⊆ 𝑦 → ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
44	29, 43	sylbi 218	. . . . 5 ⊢ (𝑧 ∈ 𝒫 𝑦 → ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
45	44	ralbiia 3084	. . . 4 ⊢ (∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
46	2, 28, 45	3anbi123i 1161	. . 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 1855	. 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 232	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 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ⊆ wss 3890  𝒫 cpw 4536   class class class wbr 5079   ≈ cen 8887   ≼ cdom 8888   ≺ csdm 8889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-groth 10744

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-dom 8892  df-sdom 8893

This theorem is referenced by:  grothomex  10750  grothac  10751