Theorem axgroth6 10726

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 10722	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
2		biid 261	. . . 4 ⊢ (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)
3		pweq 4563	. . . . . . . . 9 ⊢ (𝑧 = 𝑣 → 𝒫 𝑧 = 𝒫 𝑣)
4	3	sseq1d 3962	. . . . . . . 8 ⊢ (𝑧 = 𝑣 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑣 ⊆ 𝑦))
5	4	cbvralvw 3211	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ↔ ∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦)
6		ssid 3953	. . . . . . . . . 10 ⊢ 𝒫 𝑧 ⊆ 𝒫 𝑧
7		sseq2 3957	. . . . . . . . . . 11 ⊢ (𝑤 = 𝒫 𝑧 → (𝒫 𝑧 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
8	7	rspcev 3573	. . . . . . . . . 10 ⊢ ((𝒫 𝑧 ∈ 𝑦 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)
9	6, 8	mpan2 691	. . . . . . . . 9 ⊢ (𝒫 𝑧 ∈ 𝑦 → ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)
10		pweq 4563	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → 𝒫 𝑣 = 𝒫 𝑤)
11	10	sseq1d 3962	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (𝒫 𝑣 ⊆ 𝑦 ↔ 𝒫 𝑤 ⊆ 𝑦))
12	11	rspccv 3570	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (𝑤 ∈ 𝑦 → 𝒫 𝑤 ⊆ 𝑦))
13		pwss 4572	. . . . . . . . . . . 12 ⊢ (𝒫 𝑤 ⊆ 𝑦 ↔ ∀𝑣(𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑦))
14		vpwex 5317	. . . . . . . . . . . . 13 ⊢ 𝒫 𝑧 ∈ V
15		sseq1 3956	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝒫 𝑧 → (𝑣 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝑤))
16		eleq1 2821	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝒫 𝑧 → (𝑣 ∈ 𝑦 ↔ 𝒫 𝑧 ∈ 𝑦))
17	15, 16	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝒫 𝑧 → ((𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑦) ↔ (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦)))
18	14, 17	spcv 3556	. . . . . . . . . . . 12 ⊢ (∀𝑣(𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑦) → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦))
19	13, 18	sylbi 217	. . . . . . . . . . 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 3132	. . . . . . . . 9 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑦))
22	9, 21	impbid2 226	. . . . . . . 8 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (𝒫 𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
23	22	ralbidv 3156	. . . . . . 7 ⊢ (∀𝑣 ∈ 𝑦 𝒫 𝑣 ⊆ 𝑦 → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 ↔ ∀𝑧 ∈ 𝑦 ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
24	5, 23	sylbi 217	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 → (∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦 ↔ ∀𝑧 ∈ 𝑦 ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤))
25	24	pm5.32i 574	. . . . 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 3093	. . . . 5 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ↔ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑧 ∈ 𝑦 𝒫 𝑧 ∈ 𝑦))
27		r19.26 3093	. . . . 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 4554	. . . . . 6 ⊢ (𝑧 ∈ 𝒫 𝑦 ↔ 𝑧 ⊆ 𝑦)
30		impexp 450	. . . . . . . . 9 ⊢ (((𝑧 ⊆ 𝑦 ∧ 𝑧 ≼ 𝑦) → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)) ↔ (𝑧 ⊆ 𝑦 → (𝑧 ≼ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦))))
31		ssdomg 8929	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑧 ⊆ 𝑦 → 𝑧 ≼ 𝑦))
32	31	elv 3442	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝑦 → 𝑧 ≼ 𝑦)
33	32	pm4.71i 559	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ⊆ 𝑦 ∧ 𝑧 ≼ 𝑦))
34	33	imbi1i 349	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)) ↔ ((𝑧 ⊆ 𝑦 ∧ 𝑧 ≼ 𝑦) → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
35		brsdom 8903	. . . . . . . . . . . 12 ⊢ (𝑧 ≺ 𝑦 ↔ (𝑧 ≼ 𝑦 ∧ ¬ 𝑧 ≈ 𝑦))
36	35	imbi1i 349	. . . . . . . . . . 11 ⊢ ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ ((𝑧 ≼ 𝑦 ∧ ¬ 𝑧 ≈ 𝑦) → 𝑧 ∈ 𝑦))
37		impexp 450	. . . . . . . . . . 11 ⊢ (((𝑧 ≼ 𝑦 ∧ ¬ 𝑧 ≈ 𝑦) → 𝑧 ∈ 𝑦) ↔ (𝑧 ≼ 𝑦 → (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
38	36, 37	bitri 275	. . . . . . . . . 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 272	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑦 → ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ≈ 𝑦 → 𝑧 ∈ 𝑦)))
42		pm4.64 849	. . . . . . 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 217	. . . . 5 ⊢ (𝑧 ∈ 𝒫 𝑦 → ((𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
45	44	ralbiia 3077	. . . 4 ⊢ (∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
46	2, 28, 45	3anbi123i 1155	. . 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 1849	. 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 231	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ⊆ wss 3898  𝒫 cpw 4549   class class class wbr 5093   ≈ cen 8872   ≼ cdom 8873   ≺ csdm 8874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-groth 10721

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-dom 8877  df-sdom 8878

This theorem is referenced by:  grothomex  10727  grothac  10728