Theorem grothpw 10823

Description: Derive the Axiom of Power Sets ax-pow 5356 from the Tarski-Grothendieck axiom ax-groth 10820. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10820. Note that ax-pow 5356 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 482	. . . . . . . 8 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → 𝒫 𝑧 ⊆ 𝑦)
2	1	ralimi 3077	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦)
3		pweq 4611	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
4	3	sseq1d 4008	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑥 ⊆ 𝑦))
5	4	rspccv 3603	. . . . . . 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 616	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)) → (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)))
8	7	3adant3 1129	. . . 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		pm3.35 800	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)) → 𝒫 𝑥 ⊆ 𝑦)
10		vex 3472	. . . . 5 ⊢ 𝑦 ∈ V
11	10	ssex 5314	. . . 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 10821	. . 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 1926	. 2 ⊢ 𝒫 𝑥 ∈ V
15		axpweq 5341	. 2 ⊢ (𝒫 𝑥 ∈ V ↔ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
16	14, 15	mpbi 229	1 ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084  ∀wal 1531  ∃wex 1773   ∈ wcel 2098  ∀wral 3055  ∃wrex 3064  Vcvv 3468   ⊆ wss 3943  𝒫 cpw 4597   class class class wbr 5141   ≈ cen 8938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-groth 10820

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599

This theorem is referenced by: (None)