Theorem axgroth5 10245

Description: The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)

Assertion

Ref	Expression
axgroth5	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem axgroth5

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-groth 10244	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
2		biid 263	. . . 4 ⊢ (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)
3		pwss 4563	. . . . . 6 ⊢ (𝒫 𝑧 ⊆ 𝑦 ↔ ∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦))
4		pwss 4563	. . . . . . 7 ⊢ (𝒫 𝑧 ⊆ 𝑤 ↔ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤))
5	4	rexbii 3247	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤))
6	3, 5	anbi12i 628	. . . . 5 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
7	6	ralbii 3165	. . . 4 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
8		df-ral 3143	. . . . 5 ⊢ (∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
9		velpw 4543	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 𝑦 ↔ 𝑧 ⊆ 𝑦)
10	9	imbi1i 352	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) ↔ (𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
11	10	albii 1816	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) ↔ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
12	8, 11	bitri 277	. . . 4 ⊢ (∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
13	2, 7, 12	3anbi123i 1151	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))))
14	13	exbii 1844	. 2 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))))
15	1, 14	mpbir 233	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   ∧ w3a 1083  ∀wal 1531  ∃wex 1776   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  𝒫 cpw 4538   class class class wbr 5065   ≈ cen 8505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-groth 10244

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3942  df-ss 3951  df-pw 4540

This theorem is referenced by:  grothpw  10247  grothpwex  10248  axgroth6  10249  grothtsk  10256