Theorem grothpw 9527

Description: Derive the Axiom of Power Sets ax-pow 4769 from the Tarski-Grothendieck axiom ax-groth 9524. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4769 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
grothpw	⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem grothpw

Step	Hyp	Ref	Expression
1		simpl 472	. . . . . . . 8 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → 𝒫 𝑧 ⊆ 𝑦)
2	1	ralimi 2936	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦)
3		pweq 4111	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
4	3	sseq1d 3595	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑥 ⊆ 𝑦))
5	4	rspccv 3279	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 → (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))
6	2, 5	syl 17	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))
7	6	anim2i 591	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)) → (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)))
8	7	3adant3 1074	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) → (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)))
9		pm3.35 609	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)) → 𝒫 𝑥 ⊆ 𝑦)
10		vex 3176	. . . . 5 ⊢ 𝑦 ∈ V
11	10	ssex 4730	. . . 4 ⊢ (𝒫 𝑥 ⊆ 𝑦 → 𝒫 𝑥 ∈ V)
12	8, 9, 11	3syl 18	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) → 𝒫 𝑥 ∈ V)
13		axgroth5 9525	. . 3 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
14	12, 13	exlimiiv 1846	. 2 ⊢ 𝒫 𝑥 ∈ V
15		pwidg 4121	. . . . 5 ⊢ (𝒫 𝑥 ∈ V → 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥)
16		pweq 4111	. . . . . . 7 ⊢ (𝑦 = 𝒫 𝑥 → 𝒫 𝑦 = 𝒫 𝒫 𝑥)
17	16	eleq2d 2673	. . . . . 6 ⊢ (𝑦 = 𝒫 𝑥 → (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥 ∈ 𝒫 𝒫 𝑥))
18	17	spcegv 3267	. . . . 5 ⊢ (𝒫 𝑥 ∈ V → (𝒫 𝑥 ∈ 𝒫 𝒫 𝑥 → ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦))
19	15, 18	mpd 15	. . . 4 ⊢ (𝒫 𝑥 ∈ V → ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦)
20		elex 3185	. . . . 5 ⊢ (𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
21	20	exlimiv 1845	. . . 4 ⊢ (∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦 → 𝒫 𝑥 ∈ V)
22	19, 21	impbii 198	. . 3 ⊢ (𝒫 𝑥 ∈ V ↔ ∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦)
23	10	elpw2 4755	. . . . 5 ⊢ (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ 𝒫 𝑥 ⊆ 𝑦)
24		pwss 4123	. . . . . 6 ⊢ (𝒫 𝑥 ⊆ 𝑦 ↔ ∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦))
25		dfss2 3557	. . . . . . . 8 ⊢ (𝑧 ⊆ 𝑥 ↔ ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
26	25	imbi1i 338	. . . . . . 7 ⊢ ((𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) ↔ (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
27	26	albii 1737	. . . . . 6 ⊢ (∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
28	24, 27	bitri 263	. . . . 5 ⊢ (𝒫 𝑥 ⊆ 𝑦 ↔ ∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
29	23, 28	bitri 263	. . . 4 ⊢ (𝒫 𝑥 ∈ 𝒫 𝑦 ↔ ∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
30	29	exbii 1764	. . 3 ⊢ (∃𝑦𝒫 𝑥 ∈ 𝒫 𝑦 ↔ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
31	22, 30	bitri 263	. 2 ⊢ (𝒫 𝑥 ∈ V ↔ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
32	14, 31	mpbi 219	1 ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-groth 9524

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110

This theorem is referenced by: (None)