Theorem grothpwex 10841

Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10837. Note that ax-pow 5335 is not used by the proof. Use axpweq 5321 to obtain ax-pow 5335. Use pwex 5350 or pwexg 5348 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
grothpwex	⊢ 𝒫 𝑥 ∈ V

Proof of Theorem grothpwex

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . 7 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → 𝒫 𝑧 ⊆ 𝑦)
2	1	ralimi 3073	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦)
3		pweq 4589	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥)
4	3	sseq1d 3990	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑥 ⊆ 𝑦))
5	4	rspccv 3598	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 → (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))
6	2, 5	syl 17	. . . . 5 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))
7	6	anim2i 617	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)) → (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)))
8	7	3adant3 1132	. . 3 ⊢ ((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		pm3.35 802	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)) → 𝒫 𝑥 ⊆ 𝑦)
10		vex 3463	. . . 4 ⊢ 𝑦 ∈ V
11	10	ssex 5291	. . 3 ⊢ (𝒫 𝑥 ⊆ 𝑦 → 𝒫 𝑥 ∈ V)
12	8, 9, 11	3syl 18	. 2 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) → 𝒫 𝑥 ∈ V)
13		axgroth5 10838	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
14	12, 13	exlimiiv 1931	1 ⊢ 𝒫 𝑥 ∈ V

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ⊆ wss 3926  𝒫 cpw 4575   class class class wbr 5119   ≈ cen 8956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-groth 10837

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-in 3933  df-ss 3943  df-pw 4577

This theorem is referenced by: (None)