Theorem grothpw 10938

Description: Derive the Axiom of Power Sets ax-pow 5331 from the Tarski-Grothendieck axiom ax-groth 10935. That it follows is mentioned by Bob Solovay at https://fomarchive.ugent.be/2008-March/012783.html 10935. Note that ax-pow 5331 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 488	. . . . . . . 8 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → 𝒫 𝑧 ⊆ 𝑦)
2	1	ralimi 3103	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦)
3		pweq 4574	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
4	3	sseq1d 3944	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑥 ⊆ 𝑦))
5	4	rspccv 3577	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 → (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))
6	2, 5	syl 18	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))
7	6	anim2i 629	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)) → (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)))
8	7	3adant3 1150	. . . 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		pm3.35 815	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)) → 𝒫 𝑥 ⊆ 𝑦)
10		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
11	10	ssex 5286	. . . 4 ⊢ (𝒫 𝑥 ⊆ 𝑦 → 𝒫 𝑥 ∈ V)
12	8, 9, 11	3syl 19	. . 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 10936	. . 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 1966	. 2 ⊢ 𝒫 𝑥 ∈ V
15		axpweq 5316	. 2 ⊢ (𝒫 𝑥 ∈ V ↔ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
16	14, 15	mpbi 233	1 ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560   class class class wbr 5106   ≈ cen 8998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-groth 10935

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562

This theorem is used by: (None)